Macdonald polynomials form a two parameter basis for the ring of symmetric functions and have a very rich structure. For example, (nonsymmetric) Macdonald polynomials can be understood as eigenvectors of certain operators. In special limits they relate to Demazure characters. Recently, their structure has also been related to crystal bases, which originally came from the representation theory of quantum groups. This course will investigate these exciting new connections!

- Symmetric functions, in particular Macdonald polynomials
- Combinatorics of Coxeter groups, weak and strong Bruhat order, quantum Bruhat graph
- Crystal graphs, Demazure crystals
- Models for crystals

## Polynomials and the mod 2 Steenrod Algebra

##### This book has been cited by the following publications. This list is generated based on data provided by CrossRef.

- Publisher: Cambridge University Press
- Online publication date: November 2017
- Print publication year: 2017
- Online ISBN: 9781108304092
- DOI: https://doi.org/10.1017/9781108304092

- Subjects: Mathematics (general), Mathematics, Discrete Mathematics Information Theory and Coding, Geometry and Topology
- Series: London Mathematical Society Lecture Note Series (442)

Email your librarian or administrator to recommend adding this book to your organisation's collection.

### Book description

This is the first book to link the mod 2 Steenrod algebra, a classical object of study in algebraic topology, with modular representations of matrix groups over the field F of two elements. The link is provided through a detailed study of Peterson's `hit problem' concerning the action of the Steenrod algebra on polynomials, which remains unsolved except in special cases. The topics range from decompositions of integers as sums of 'powers of 2 minus 1', to Hopf algebras and the Steinberg representation of GL(n, F). Volume 1 develops the structure of the Steenrod algebra from an algebraic viewpoint and can be used as a graduate-level textbook. Volume 2 broadens the discussion to include modular representations of matrix groups.

### Reviews

'In these volumes, the authors draw upon the work of many researchers in addition to their own work, in places presenting new proofs or improvements of results. Moreover, the material in Volume 2 using the cyclic splitting of P(n) is based in part upon the unpublished Ph.D. thesis of Helen Weaver … Much of the material covered has not hitherto appeared in book form, and these volumes should serve as a useful reference. … readers will find different aspects appealing.'

Geoffrey M. L. Powell Source: Mathematical Reviews

##### Refine List

## Grothendieck rings of basic classical Lie superalgebras

The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corresponding generalized root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called super Weyl groupoids.

[Brun] J. Brundan, "Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $mathfrak*J. Amer. Math. Soc.*, vol. 16, iss. 1, pp. 185-231, 2003. [BK] J. Brundan and J. Kujawa, "A new proof of the Mullineux conjecture,"

*J. Algebraic Combin.*, vol. 18, iss. 1, pp. 13-39, 2003. [CV1990] O. A. Chalykh and A. P. Veselov, "Commutative rings of partial differential operators and Lie algebras,"

*Commun. Math. Phys.*, vol. 126, iss. 3, pp. 597-611, 1990. [Hecken1] I. Heckenberger, "The Weyl groupoid of a Nichols algebra of diagonal type,"

*Invent. Math.*, vol. 164, iss. 1, pp. 175-188, 2006. [Hecken2] I. Heckenberger and H. Yamane, "A generalization of Coxeter groups, root systems, and Matsumoto’s theorem,"

*Math. Z.*, vol. 259, iss. 2, pp. 255-276, 2008. [Kac2] V. G. Kac, "Representations of classical Lie superalgebras," in

*Differential Geometrical Methods in Mathematical Physics, II*, New York: Springer-Verlag, 1978, vol. 676, pp. 597-626. [Kac3] V. G. Kac, "Laplace operators of infinite-dimensional Lie algebras and theta functions,"

*Proc. Nat. Acad. Sci. U.S.A.*, vol. 81, iss. 2, Phys. Sci., pp. 645-647, 1984. [KV] H. M. Khudaverdian and T. T. Voronov, "Berezinians, exterior powers and recurrent sequences,"

*Lett. Math. Phys.*, vol. 74, iss. 2, pp. 201-228, 2005. [Serga1] V. Serganova, "Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra $mathfrak

*Selecta Math.*, vol. 2, iss. 4, pp. 607-651, 1996. [Serga4] V. Serganova, "Kac-Moody superalgebras and integrability," in

*Developments and Trends in Infinite-Dimensional Lie Theory*, Boston, MA: Birkhäuser Boston, Inc., 2011, vol. 288, pp. 169-218. [Serge1] A. Sergeev, "The invariant polynomials on simple Lie superalgebras,"

*Represent. Theory*, vol. 3, pp. 250-280, 1999. [SV] A. Sergeev and A. P. Veselov, "Deformed quantum Calogero-Moser problems and Lie superalgebras,"

*Commun. Math. Phys.*, vol. 245, iss. 2, pp. 249-278, 2004. [SV1] A. Sergeev and A. P. Veselov, "Deformed Macdonald-Ruijsenaars operators and super Macdonald polynomials,"

*Commun. Math. Phys.*, vol. 288, iss. 2, pp. 653-675, 2009.

## UCD MAT 280: Macdonald Polynomials and Crystal Bases - Mathematics

Today, November 30 th , is AMS Day! Join our celebration of AMS members and explore special offers on AMS publications, membership and more. Offers end 11:59pm EST.

ISSN 1088-6834(online) ISSN 0894-0347(print)

Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $mathfrak

Author: Jonathan Brundan

Journal: J. Amer. Math. Soc. **16** (2003), 185-231

MSC (2000): Primary 17B10

DOI: https://doi.org/10.1090/S0894-0347-02-00408-3

Published electronically: October 16, 2002

MathSciNet review: 1937204

Full-text PDF Free Access

Abstract: We compute the characters of the finite dimensional irreducible representations of the Lie superalgebra $mathfrak

- Alexandre Beĭlinson and Joseph Bernstein,
*Localisation de $g$-modules*, C. R. Acad. Sci. Paris Sér. I Math.**292**(1981), no. 1, 15–18 (French, with English summary). MR**610137** - Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel,
*Koszul duality patterns in representation theory*, J. Amer. Math. Soc.**9**(1996), no. 2, 473–527. MR**1322847**, DOI https://doi.org/10.1090/S0894-0347-96-00192-0 [BR]BR A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras,*Advances Math.***64**(1987), 118–175. - I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand,
*A certain category of $*, Funkcional. Anal. i Priložen.$-modules **10**(1976), no. 2, 1–8 (Russian). MR**0407097** - I. N. Bernšteĭn and D. A. Leĭtes,
*A formula for the characters of the irreducible finite-dimensional representations of Lie superalgebras of series $< m Gl>$ and $< m sl>$*, C. R. Acad. Bulgare Sci.**33**(1980), no. 8, 1049–1051 (Russian). MR**620836** - Nicolas Bourbaki,
*Commutative algebra. Chapters 1–7*, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French Reprint of the 1972 edition. MR**979760** - Jonathan Brundan,
*Modular branching rules and the Mullineux map for Hecke algebras of type $A$*, Proc. London Math. Soc. (3)**77**(1998), no. 3, 551–581. MR**1643413**, DOI https://doi.org/10.1112/S0024611598000562 [B2]Btilt J. Brundan, Tilting modules for Lie superalgebras, preprint, University of Oregon, 2002, available from http://darkwing.uoregon.edu/

- [BB]BB A. Beilinson and J. Bernstein, Localisation de $mathfrak g$-modules,

*C. R. Acad. Sci. Paris Ser. I Math.*

**292**(1981), 15–18. [BGS]BGS A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory,

*J. Amer. Math. Soc.*

**9**(1996), 473–527. [BR]BR A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras,

*Advances Math.*

**64**(1987), 118–175. [BGG]BGG J. Bernstein, I. M. Gelfand and S. I. Gelfand, A category of $mathfrak g$-modules,

*Func. Anal. Appl.*

**10**(1976), 87–92. [BL]BL J. Bernstein and D. Leites, Character formulae for irreducible representations of Lie superalgebras of series $mathfrak

*C. R. Acad. Bulg. Sci.*

**33**(1980), 1049–1051. [Bou]BouCA N. Bourbaki,

*Commutative algebra*, Springer, 1989. [B1]JWB:branching J. Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type $mathbf A$,

*Proc. London Math. Soc.*

**77**(1998), 551–581. [B2]Btilt J. Brundan, Tilting modules for Lie superalgebras, preprint, University of Oregon, 2002, available from http://darkwing.uoregon.edu/

Retrieve articles in *Journal of the American Mathematical Society* with MSC (2000): 17B10

Retrieve articles in all journals with MSC (2000): 17B10

**Jonathan Brundan**

Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Email: [email protected]

Received by editor(s): March 12, 2002

Received by editor(s) in revised form: September 25, 2002

Published electronically: October 16, 2002

Additional Notes: Research partially supported by the NSF (grant no. DMS-0139019)

Article copyright: © Copyright 2002 American Mathematical Society

## UCD MAT 280: Macdonald Polynomials and Crystal Bases - Mathematics

8 papers (marked by #) are not in MathSciNet.

1. Model de geometrie afina plana peste un corp finit, Studii Cerc.Mat.17(1965), 1337-1340.

2. Constructia fibrarilor universale peste poliedre arbitrare, Studii Cerc.Mat. 18(1965), 1215-1219.

3. (with H.Moscovici) Démonstration du théorème sur la suite spectrale d'un fibré au sens de Kan, Proc.Camb.Phil.Soc.64(1968), 293-297.

4. Sur les complexes elliptiques fibrés, C.R.Acad.Sci.Paris(A)266(1968), 914-917.

5. Sur les actions libres des groupes finis, Bull.Acad.Polon.Sci.16(1968), 461-463.

6. Coomologia complexelor eliptice, Studii Cerc.Mat.21(1969), 38-83.

7. A property of certain non-degenerate holomorphic vector fields, An.Univ.Timisoara 7(1969), 73-76.

8. (with J.Milnor and F.P.Peterson) Semicharacteristics and cobordism, Topology 8(1969), 357-359.

9. Remarks on the holomorphic Lefschetz formula, in "Analyse globale", Presses de l'Univ. de Montréal 1969, 193-204.

10. (with J.Dupont) On manifolds satisfying w_1^2=0, Topology 10(1971), 81-92.

11. Novikov's higher signature and families of elliptic operators, J.Diff.Geom. 7(1972), 229-256.

12. On the discrete series representations of the general linear groups over a finite field, Bull.Amer.Math.Soc. 79(1973), 550-554.

13. THE DISCRETE SERIES OF GL_n OVER A FINITE FIELD, Ann.Math. Studies 81, Princeton U.Press 1974, 99p.

14. Introduction to elliptic operators, in "Global Analysis and applications", Internat.Atomic Energy Agency, Vienna 1974, 187-193.

15. (with R.W.Carter) On the modular representations of the general linear and symmetric groups, Math.Z.136(1974), 193-242.

16. (with R.W.Carter) Modular representations of the general linear and symmetric groups, Proc.2nd Int.Conf.Th.Groups 1973, LNM 372, Springer Verlag 1974, 218-220.

17. On the discrete series representations of the classical groups over a finite field, in "Proc.Int.Congr.Math.,Vancouver 1974", 1975, 465-470.

18. Sur la conjecture de Macdonald, C.R.Acad.Sci.Paris(A) 280(1975), 371-320.

19. Divisibility of projective modules of finite Chevalley groups by the Steinberg module, Bull.Lond.Math.Soc.8 (1976), 130-134.

20. A note on counting nilpotent matrices of fixed rank, Bull.Lond.Math.Soc.8(1976), 77-80.

21. (with R.W.Carter) Modular representations of finite groups of Lie type, Proc.Lond.Math.Soc.32(1976), 347-384.

22. (with P.Deligne) Representations of reductive groups over finite fields, Ann.Math.103(1976), 103-161.

23. On the Green polynomials of classical groups, Proc.Lond.Math.Soc.33(1976), 443-475.

24. On the finiteness of the number of unipotent classes, Invent.Math. 34(1976), 201-213.

25. Coxeter orbits and eigenspaces of Frobenius, Invent.Math.28(1976), 101-159.

26. (with J.A.Green and G.I.Lehrer) On the degrees of certain group characters, Quart.J.Math.27(1976), 1-4.

27. (with B.Srinivasan) The characters of the finite unitary groups, J.Algebra 49(1977), 167-171.

28. Classification des représentations irréductibles des groupes classiques finis, C.R.Acad.Sci.Paris(A) 284(1977), 473-476.

29. Irreducible representations of finite classical groups, Invent.Math.43(1977), 125-175.

30. Representations of finite Chevalley groups, Regional Conf. Series in Math.39, Amer.Math.Soc. 1978, 48p. New errata

31. (with W.M.Beynon) Some numerical results on the characters of exceptional Weyl groups, Math.Proc.Camb.Phil.Soc.84(1978), 417-426.

32. Some remarks on the supercuspidal representations of p-adic semisimple groups, Proc.Symp.Pure Math.33(1), Amer.Math.Soc. 1979, 171-175.

33.# On the reflection representation of a finite Chevalley group, in "Representation theory of Lie groups", LMS Lect.Notes Ser.34, Cambridge U.Press 1979, 325-337.

34. Unipotent representations of a finite Chevalley group of type E_8, Quart.J.Math.30(1979), 315-338.

35. (with N.Spaltenstein) Induced unipotent classes, J.Lond.Math.Soc.19(1979), 41-52.

36. A class of irreducible representations of a Weyl group, Proc.Kon.Nederl.Akad.(A) 82(1979), 323-335.

37. (with D.Kazhdan) Representations of Coxeter groups and Hecke algebras, Invent.Math.53(1979), 165-184.

38. (with D.Kazhdan) A topological approach to Springer's representations, Adv.Math.38(1980), 222-228.

39. (with D.Kazhdan) Schubert varieties and Poincaré duality, Proc.Symp.Pure Math.36, Amer.Math.Soc. 1980, 185-203.

40. Some problems in the representation theory of finite Chevalley groups, Proc.Symp.Pure Math.37, Amer.Math.Soc. 1980, 313-317.

41. Hecke algebras and Jantzen's generic decomposition patterns, Adv.Math.37(1980), 121-164.

42. On the unipotent characters of the exceptional groups over finite fields, Invent.Math.60(1980), 173-192.

43. On a theorem of Benson and Curtis, J.Algebra 71(1981), 490-498.

44. Green polynomials and singularities of unipotent classes, Adv.Math.42(1981), 169-178.

45. Unipotent characters of the symplectic and odd orthogonal groups over a finite field, Invent.Math.64(1981), 263-296.

46. Unipotent characters of the even orthogonal groups over a finite field, Trans.Amer.Math.Soc.272(1982), 733-751.

47. (with P.Deligne) Duality for representations of a reductive group over a finite field, J.Algebra 74(1982), 284-291.

48. (with D.Alvis) On Springer's correspondence for simple groups of type E_n (n=6,7,8), Math.Proc.Camb.Phil.Soc. 92(1982), 65-72.

49. (with D.Alvis) The representations and generic degrees of the Hecke algebras of type H_4, J. für reine und angew.math.336(1982), 201-212 Erratum, 449(1994), 217-218.

50. A class of irreducible representations of a Weyl group II, Proc.Kon.Nederl.Akad.(A) 85(1982), 219-226.

51. (with D.Vogan) Singularities of closures of K-orbits on a flag manifold, Invent.Math.71(1983), 365-379.

52. (with P.Deligne) Duality for representations of a reductive group over a finite field II, J.Algebra 81(1983), 540-549.

54. Some examples of square integrable representations of semisimple p-adic groups, Trans.Amer.Math.Soc.227(1983), 623-653.

55. Left cells in Weyl groups, in "Lie groups representations", LNM 1024, Springer Verlag 1983, 99-111.

56.# Open problems in algebraic groups, Proc.12th Int.Symp., Taniguchi Foundation, Katata 1983, p.14.

57. CHARACTERS OF REDUCTIVE GROUPS OVER A FINITE FIELD, Ann.Math.Studies 107, Princeton U.Press 1984, 384p. Errata (ps)

58. Characters of reductive groups over finite fields, in "Proc.Int.Congr.Math., Warsaw 1983", North Holland 1984 877-880.

59. Intersection cohomology complexes on a reductive group, Invent.Math.75(1984), 205-272.

60. Cells in affine Weyl groups, in "Algebraic groups and related topics", Adv.Stud.Pure Math.6, North-Holland and Kinokuniya 1985, 255-287.

61. (with N.Spaltenstein) On the generalized Springer correspondence for classical groups, in "Algebraic groups and related topics", Adv.Stud.Pure Math.6, North Holland and Kinokuniya 1985, pp. 289-316.

62. The two sided cells of the affine Weyl group of type A, in "Infinite dimensional groups with applications", MSRI Publ.4, Springer Verlag 1985, pp.275-283.

63. Character sheaves I, Adv.Math.56(1985), 193-237.

64. Character sheaves II, Adv.Math.57(1985), 226-265.

65. Character sheaves III, Adv.Math.57(1985), 266-315.

66. Equivariant K-theory and representations of Hecke algebras, Proc.Amer.Math.Soc. 94(1985), 337-342.

67. (with D.Kazhdan) Equivariant K-theory and representations of Hecke algebras II, Invent.Math.80(1985), 209-231.

68. Character sheaves IV, Adv.Math.59(1986), 1-63.

69. Character sheaves V, Adv.Math.61(1986), 103-155.

70. Sur les cellules gauches des groupes de Weyl, C.R.Acad.Sci.Paris(A), 302(1986), 5-8.

71. On the character values of finite Chevalley groups at unipotent elements, J.Algebra 104(1986), 146-194.

72. (with D.Kazhdan) Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent.Math.87(1987), 153-215.

73. Cells in affine Weyl groups II, J.Algebra 109(1987), 536-548.

74. Fourier transforms on a semisimple Lie algebra over F_q, in "Algebraic Groups-Utrecht 1986", LNM 1271, Springer Verlag 1987, 177-188.

75. Cells in affine Weyl groups III, J.Fac.Sci.Tokyo U.(IA) 34(1987), 223-243.

76. Introduction to character sheaves, Proc.Symp.Pure Math.47(1), Amer.Math.Soc. 1987, 165-180.

77. Leading coefficients of character values of Hecke algebras, Proc.Symp.Pure Math.47(2), Amer.Math.Soc. 1987, 235-262.

78. (with C.De Concini and C.Procesi) Homology of the zero set of a nilpotent vector field on a flag manifold, J.Amer.Math.Soc.1(1988), 15-34.

79. Quantum deformations of certain simple modules over enveloping algebras, Adv.Math.70(1988), 237-249.

80. (with D.Kazhdan) Fixed point varieties on affine flag manifolds, Isr.J.Math.62(1988), 129-168.

81. Cuspidal local systems and graded Hecke algebras I, Publ.Math.I.H.E.S.67(1988), 145-202.

82. (with N.Xi) Canonical left cells in affine Weyl groups, Adv.Math.72(1988), 284-288.

83. On representations of reductive groups with disconnected center, Astérisque 168(1988), 157-166.

84. Modular representations and quantum groups, Contemp.Math.82(1989), 59-77.

85. Affine Hecke algebras and their graded version, J.Amer.Math.Soc.2(1989), 599-635.

86. Cells in affine Weyl groups IV, J.Fac.Sci.Tokyo U.(IA) 36(1989), 297-328.

87. Representations of affine Hecke algebras, Astérisque 171-172(1989), 73-84.

88. On quantum groups, J.Algebra 131(1990), 466-475.

89. Green functions and character sheaves, Ann.Math. 131(1990), 355-408.

90. Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J.Amer.Math.Soc. 3(1990), 257-296.

91. Quantum groups at roots of 1, Geom.Ded.35(1990), 89-114.

92. Canonical bases arising from quantized enveloping algebras, J.Amer.Math.Soc.3(1990), 447-498.

93. (with A.A.Beilinson and R.MacPherson) A geometric setting for the quantum deformation of GL_n, Duke Math.J. 61(1990), 655-677.

94. Symmetric spaces over a finite field, in "The Grothendieck Festschrift III", Progr.in Math.88, Birkhauser Boston 1990, 57-81. Errata (ps)

95. Canonical bases arising from quantized enveloping algebras II (ps), in "Common trends in mathematics and quantum field theories", Progr.of Theor.Phys.Suppl.102(1990), ed. T.Eguchi&al., 175-201.

96. (with D.Kazhdan) Affine Lie algebras and quantum groups, Int.Math.Res.Notices 1991, 21-29.

97. Quivers, perverse sheaves and enveloping algebras, J.Amer.Math.Soc.4(1991), 365-421.

98. (with J.M.Smelt) Fixed point varieties in the space of lattices, Bull.Lond.Math.Soc.23(1991), 213-218.

99. Intersection cohomology methods in representation theory, in "Proc.Int.Congr.Math.Kyoto 1990", Springer Verlag 1991, 155-174.

100. A unipotent support for irreducible representations, Adv.Math.94(1992), 139-179.

101. Canonical bases in tensor products, Proc.Nat.Acad.Sci.89(1992), 8177-8179.

102. Remarks on computing irreducible characters, J.Amer.Math.Soc.5(1992), 971-986.

103. Introduction to quantized enveloping algebras, in Progr.in Math.105, Birkhauser Boston 1992, 49-65.

104. Affine quivers and canonical bases, Publ.Math.I.H.E.S.76(1992), 111-163.

105. (with J.Tits) The inverse of a Cartan matrix (ps), An.Univ.Timisoara 30(1992), 17-23.

106. (with I.Grojnowski) On bases of irreducible representations of quantum GL_n, in "Kazhdan-Lusztig theory and related topics", Contemp.Math.139(1992), 167-174.

107. INTRODUCTION TO QUANTUM GROUPS, Progr.in Math.110, Birkhauser Boston 1993, 341p. (Reprinted 1994,2010.) Errata(4/2010) (ps) [From a mathoverflow.net contributor: "don't be fooled by the title!"]

108. (with D.Kazhdan) Tensor structures arising from affine Lie algebras I, J.Amer.Math.Soc.6(1993), 905-947.

109. (with D.Kazhdan) Tensor structures arising from affine Lie algebras II, J.Amer.Math.Soc.6(1993), 949-1011.

110. Coxeter groups and unipotent representations, Astérisque 212(1993), 191-203.

111. (with I.Grojnowski) A comparison of bases of quantized enveloping algebras, in "Linear algebraic groups and their representations", Contemp.Math.153(1993), 11-19.

112. Tight monomials in quantized enveloping algebras,(ps) in "Quantum deformations of algebras and their representations" ed. A.Joseph&al., Isr.Math.Conf.Proc.7(1993), Amer.Math.Soc. 117-132.

113. Exotic Fourier transform, Duke Math.J.73(1994), 227-241.

114. Vanishing properties of cuspidal local systems, Proc.Nat.Acad.Sci.91(1994), 1438-1439.

115. (with D.Kazhdan) Tensor structures arising from affine Lie algebras III, J.Amer.Math.Soc.7(1994), 335-381.

116. (with D.Kazhdan) Tensor structures arising from affine Lie algebras IV, J.Amer.Math.Soc.7(1994), 383-453.

117. Monodromic systems on affine flag manifolds, Proc.Roy.Soc.Lond.(A) 445(1994), 231-246 Errata (ps), 450(1995), 731-732.

118. Problems on canonical bases, in "Algebraic groups and their generalizations: quantum and infinite dimensional methods", Proc.Symp.Pure Math.56(2), Amer.Math.Soc. 1994, 169-176.

119. Total positivity in reductive groups, in "Lie theory and geometry", Progr.in Math.123, Birkhauser Boston 1994, 531-568. Errata

120. Study of perverse sheaves arising from graded Lie algebras, Adv.Math.112(1995), 147-217.

121. Cuspidal local systems and graded Hecke algebras II, in "Representations of groups" ed.B.Allison&al., Canad.Math.Soc.Conf.Proc.16, Amer.Math.Soc. 1995, 217-275.

122. Quantum groups at v=infinity, in "Functional analysis on the eve of the 21st century", vol.I, Progr.in Math. 131, Birkhauser Boston 1995, 199-221.

123. Classification of unipotent representations of simple p-adic groups, Int.Math.Res.Notices (1995), 517-589.

124. An algebraic-geometric parametrization of the canonical basis, Adv.Math.120(1996), 173-190.

125. Affine Weyl groups and conjugacy classes in Weyl groups, Transform.Groups 1(1996), 83-97.

126. Braid group actions and canonical bases, Adv.Math.122(1996), 237-261.

127. Non local finiteness of a W-graph, Represent.Th.1(1997), 25-30 .

128. Cohomology of classifying spaces and hermitian representations, Represent.Th.1(1997), 31-36.

129. (with C.K.Fan) Factorization of certain exponentials in Lie groups, in "Algebraic groups and Lie groups", ed. G.I.Lehrer, Cambridge U.Press 1997, 215-218.

130. Total positivity and canonical bases, in "Algebraic groups and Lie groups" ed. G.I.Lehrer, Cambridge U.Press 1997, 281-295.

131. Notes on unipotent classes, Asian J.Math.1(1997), 194-207.

132. Cells in affine Weyl groups and tensor categories, Adv.Math.129(1997), 85-98.

133. Periodic W-graphs, Represent.Th.1(1997), 207-279.

134.# A comparison of two graphs, Int.Math.Res.Notices (1997), 639-640.

135. Constructible functions on the Steinberg variety, Adv.Math.130(1997), 287-310.

136. Total positivity in partial flag manifolds, Represent.Th.2(1998), 70-78.

137. Introduction to total positivity (ps), in "Positivity in Lie theory: open problems" ed. J.Hilgert&al., de Gruyter 1998, 133-145.

138. On quiver varieties, Adv.Math.136(1998), 141-182.

139. Canonical bases and Hall algebras, in "Representation Theories and Algebraic Geometry", ed. A.Broer&al., Kluwer Acad.Publ. 1998, 365-399.

140. Bases in equivariant K-theory, Represent.Th.2(1998), 298-369.

141. Homology bases arising from reductive groups over a finite field, in "Algebraic groups and their representations", ed. R.W.Carter&al., Kluwer Acad.Publ. 1998, 53-72.

143. Bases in equivariant K-theory II, Represent.Th.3(1999), 281-353.

144. A survey of group representations, Nieuw Archief voor Wiskunde 17(1999), 483-489.

145. Subregular nilpotent elements and bases in K-theory, Canad.J.Math.51(1999), 1194-1225.

146. Recollections about my teacher, Michael Atiyah, Asian J.Math.3(1999),v-vi.

147. Semicanonical bases arising from enveloping algebras, Adv.Math.151(2000), 129-139.

149. Quiver varieties and Weyl group actions, Ann.Inst.Fourier 50(2000), 461-489.

150. G(F_q)-invariants in irreducible G(F_)-modules, Represent.Th.4(2000), 448-465.

151. Remarks on quiver varieties, Duke Math.J.105(2000), 239-265.

152. Transfer maps for quantum affine sl(n) (ps), in "Representations and quantizations", ed. J.Wang&al., China Higher Education Press and Springer Verlag 2000, 341-356.

153. Representation theory in characteristic p, in "Taniguchi Conf. on Math. Nara'98", Adv.Stud.Pure Math.31, Math.Soc.Japan, 2001, 167-178.

154. Cuspidal local systems and graded Hecke algebras III, Represent.Th.6(2002), 202-242.

155. Classification of unipotent representations of simple p-adic groups II, Represent.Th.6(2002), 243-289.

156. Constructible functions on varieties attached to quivers, in "Studies in memory of I.Schur", Progress in Math.210, Birkhauser Boston 2002, 177-223.

157. Rationality properties of unipotent representations, J.Algebra 258(2002), 1-22.

158. Notes on affine Hecke algebras, in "Iwahori-Hecke algebras and their representation theory", ed. M.W.Baldoni&al., LNM 1804, Springer Verlag 2002, 71-103,

159. HECKE ALGEBRAS WITH UNEQUAL PARAMETERS, CRM Monographs Ser.18, Amer.Math.Soc. 2003, 136p additional material in version2 (2014), arxiv:math/0208154.

160. Homomorphisms of the alternating group A5 into reductive groups, J.Algebra 260(2003), 298-322.

161. Character sheaves on disconnected groups I, Represent.Th.7(2003), 374-403 Errata 8(2004), 179-179.

162. Representations of reductive groups over finite rings, Represent.Th.8(2004), 1-14.

163. Character sheaves on disconnected groups II,Represent.Th.8(2004), 72-124.

164. Character sheaves on disconnected groups III,Represent.Th.8(2004), 125-144.

165. Character sheaves on disconnected groups IV,Represent.Th.8(2004), 145-178.

166. Parabolic character sheaves I, Mosc.Math.J.4(2004), 153-179.

167. An induction theorem for Springer's representations, in "Representation theory of Algebraic Groups and Quantum Groups", Adv.Stud.Pure Math.40, Math.Soc.Japan, Kinokuniya 2004, 253-259.

168. Character sheaves on disconnected groups V, Represent.Th.8(2004), 346-376.

169. Character sheaves on disconnected groups VI, Represent.Th.8(2004), 377-413.

170. Parabolic character sheaves II, Mosc.Math.J.4(2004), 869-896.

171. Convolution of almost characters, Asian J.Math.8(2004), 769-772.

172. Character sheaves on disconnected groups VII, Represent.Th.9(2005), 209-266.

173. Unipotent elements in small characteristic, Transform.Groups 10(2005), 449-487.

174. Character sheaves and generalizations, in "The unity of mathematics", ed. P.Etingof&al., 443-455, Progress in Math.244, Birkhauser Boston 2006.

175. A q-analogue of an identity of N. Wallach, in "Studies in Lie Theory", ed.J.Bernstein&al., 405-410, Progress in Math.243, Birkhauser Boston 2006.

176. Character sheaves on disconnected groups VIII, Represent.Th.10(2006), 314-352.

177. Character sheaves on disconnected groups IX, Represent.Th.10(2006), 353-379.

178. A class of perverse sheaves on a partial flag manifold, Represent.Th.11(2007), 122-171.

179. (with X.He) Singular supports for character sheaves on a group compactification, Geom. and Funct.Analysis 17(2007), 1915-1923.

180. Irreducible representations of finite spin groups, Represent.Th.12(2008), 1-36.

181. Generic character sheaves on disconnected groups and character values, Represent.Th.12(2008), 225-235.

182. Unipotent elements in small characteristic II, Transform.Groups 13(2008), 773-797.

183. A survey of total positivity, Milan J.Math. 76(2008), 125-134.

184. Study of a Z-form of the coordinate ring of a reductive group, J.Amer.Math.Soc. 22(2009), 739-769.

185. (with S.Kumar and D.Prasad) Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups, in "Representation theory" ed.Z.Lin&al. Contemp.Math. 478 (2009), 99-101.

186. Twelve bridges from a reductive group to its Langlands dual, in "Representation theory", ed.Z.Lin &al., Contemp.Math. 478 (2009), 125-143.

187. Character sheaves on disconnected groups X, Represent.Th.13(2009), 82-140.

188. Unipotent classes and special Weyl group representations, J.Algebra 321(2009), 3418-3449.

189. Remarks on Springer's representations, Represent.Th. 13(2009), 391-400.

190. Notes on character sheaves, Moscow Math.J. 9(2009), 91-109.

191. Graded Lie algebras and intersection cohomology (ps), in: "Representation theory of algebraic groups and quantum groups", ed. A.Gyoja&al. Progr.in Math.284, Birkhauser 2010, p.191-224.

192. Unipotent elements in small characteristic IV, Transfor.Groups 15(2010), 921-936.

193. Parabolic character sheaves, III, Moscow Math.J.10(2010), 603-609.

194. Unipotent elements in small characteristic III, J.Algebra 329(2011), 163-189.

195. Piecewise linear parametrization of canonical bases, Pure Appl.Math.Quart. 7(2011),783-796.

196. On some partitions of a flag manifold, Asian J.Math. 15(2011), 1-8.

197. From conjugacy classes in the Weyl group to unipotent classes, Represent.Th. 15(2011),494-530.

198. From groups to symmetric spaces, Contemp.Math. 557(2011), 245-258.

199. Study of antiorbital complexes, Contemp.Math. 557(2011), 259-287.

200. On C-small conjugacy classes in a reductive group, Transfor.Groups, 16(2011),807-825.

202. On certain varieties attached to a Weyl group element, Bull.Inst.Math.Acad.Sinica (N.S.), 6(2011), 377-414.

203. (with X.He) A generalization of Steinberg's cross-section, J.Amer.Math.Soc. 25(2012), 739-757.

204. Elliptic elements in a Weyl group: a homogeneity property, Represent.Th. 16(2012), 127-151.

205. From conjugacy classes in the Weyl group to unipotent classes II, Represent.Th. 16(2012), 189-211.

206. On the cleanness of cuspidal character sheaves, Mosc.Math.J. 12(2012), 621-631.

207. (with T.Xue) Elliptic Weyl group elements and unipotent isometries with p=2, Represent.Th. 16(2012), 270-275.

208. (with D.Vogan) Hecke algebras and involutions in Weyl groups, Bull.Inst.Math.Acad.Sinica (N.S.) 7(2012), 323-354.

209. A bar operator for involutions in a Coxeter group, arxiv:1112.0969, Bull.Inst.Math.Acad.Sinica (N.S.) 7(2012), 355-404.

210. From conjugacy classes in the Weyl group to unipotent classes III, Represent.Th. 16(2012), 450-488.

211. On the representations of disconnected reductive groups over F_q, "Recent developments in Lie algebras, groups and representation theory", ed.K.Misra, Proc.Symp.Pure Math.86(2012), 227-246, Amer.Math.Soc.

212. (with Z.Yun) A (-q)-analogue of weight multiplicities, J.Ramanujan Math.Soc. 29A(2013), 311-340.

213. (with J.L.Kim) On the characters of unipotent representations of a semisimple p-adic group, Represent.Th.17(2013), 426-441.

214. Asymptotic Hecke algebras and involutions, in "Perspectives in Represent. Th." 267-278, ed.P.Etingof et.al., Contemp.Math. 610(2014).

215. Families and Springer's correspondence, Pacific J.Math.267(2014), 431-450.

216. Restriction of a character sheaf to conjugacy classes, Bulletin Math. Soc.Sci.Math.Roum. 58(2015), 297-309.

217. (with J.L.Kim) On the Steinberg character of a semisimple p-adic group, Pacific J.Math.265(2013), 499-509.

218. (with D.Vogan) Quasisplit Hecke algebras and symmetric spaces, Duke Math.J. 163(2014), 983-1070.

219. Unipotent almost characters of simple p-adic groups, in: De la G'eom'etrie Alg'ebrique aux Formes Automorphes, Ast'erisque 370(2015), 243-267

220. Unipotent almost characters of simple p-adic groups II, Transfor.Groups,19(2014), 527-547.

221. Distinguished conjugacy classes and elliptic Weyl group elements, Represent.Th. 18(2014), 223-277.

222. On conjugacy classes in a reductive group, Representations of Reductive Groups, Progr.in Math.312, Birkh"auser 2015, 333-363.

223. Truncated convolution of character sheaves, Bull.Inst.Math.Acad.Sinica (N.S.) 10(2015), 1-72.

224. On conjugacy classes in the Lie group E_8, Bull.Math.Soc.Math.Roumanie, 2020

225. (with D.Vogan) Hecke algebras and involutions in Coxeter groups, Representations of Reductive Groups, Progr.in Math.312, Birkh"auser 2015, 365-398.

226. Unipotent representations as a categorical centre, Represent.Th. 19(2015), 211-235.

227. Action of longest element on a Hecke algebra cell module, Pacific J.Math. 279(2015), 363-396.

228. On the character of certain irreducible modular representations, Represent.Th. 19(2015), 3-8.

229.# Algebraic and geometric methods in representation theory (Lecture at the Chinese University of Hong Kong, Sep.25, 2014), arxiv:1409.8003.

230. Some power series involving involutions in Coxeter groups, Represent.Th. 19(2015), 281-289.

231. Nonsplit Hecke algebras and perverse sheaves, Selecta Math. 22(2016), 1953-1986.

232. (with G.Williamson) On the character of certain tilting modules, Sci.China.Math 61(2018), 295-298.

233. Non-unipotent character sheaves as a categorical centre, Bull.Inst.Math.Acad.Sinica (N.S.) 11(2016), 603-731.

234. An involution based left ideal in the Hecke algebra, Represent.Th. 20(2016), 172-186.

235. Generic character sheaves on groups over $kk[e]/(e^r)$, in Categorification and higher representation theory, Contemporary Math. 683(2017), 227-246.

236. Exceptional representations of Weyl groups, J.of Algebra 475(2017), 14-20.

237. The canonical basis of the quantum adjoint representation, J.Comb.Alg. 1(2017), 45-57.

238. Generalized Springer theory and weight functions, Ann. Univ.Ferrara Sez.VII Sci.Mat. 63(2017), 159-167.

239. On the definition of almost characters, in "Lie Groups, Geometry and Representation theory" ed. V.Kac, V.Popov, Progr.in Math. 326, Birkhauser 2018.

240. Special representations of Weyl groups: a positivity property, Adv.in Math. 327(2018), 161-172.

241. (with Z. Yun) Z/m-graded Lie algebras and perverse sheaves, I, Represent.Th. 21(2017), 277-321.

242. (with Z. Yun) Z/m-graded Lie algebras and perverse sheaves, II, Represent.Th. 21(2017),322-353.

243. (with Z. Yun) Z/m-graded Lie algebras and perverse sheaves, III: graded double affine Hecke algebra, Represent.Th. 22(2018),87-118.

244. On the generalized Springer correspondence, Representations of reductive groups, Proc.Symp.Pure Math., Amer.Math.Soc., 101(2019), 219-253.

245. Non-unipotent representations and categorical centres, Bull. Inst. Math. Acad. Sinica (N.S.) 12(2017), 205-296.

246. (with G.Williamson) Billiards and tilting characters for SL_3, SIGMA Symmetry, Integrability, Geom. Methods Appl. 14(2018), 15, 22p.

247. Conjugacy classes in reductive groups and two-sided cells, Bull. Inst. Math. Acad. Sinica (N.S.) 14(2019), 265-293.

248.# Comments on my papers, arxiv:1707.09368.

249. Lifting involutions in a Weyl group to the torus normalizer, Represent.Th. 22(2018), 27-44.

250. Hecke modules based on involutions in extended Weyl groups, Represent.Th. 22(2018), 246-277.

251. Discretization of Springer fibres, arxiv:1712.07530.

252. A new basis for the representation ring of a Weyl group, Represent.Th. 23(2019), 439-461.

253. Positive conjugacy classes in Weyl groups, arxiv:1805.03772, to appear.

254. (with Z. Yun) Z/m-graded Lie algebras and perverse sheaves, IV, arxiv:1805.10550, to appear Repres.Th.

255. Reducing mod $p$ complex representations of finite reductive groups, arxiv:1810.10492.

256. Positive structures in Lie theory, arxiv:1812.09313, to appear Notices of ICCM

257. Remarks on affine Springer fibres, Bull.Inst.Math.Acad.Sinica (N.S.)2020

258. (with Z.Yun) Endoscopy for Hecke categories, character sheaves and representations, Forum of Math.Pi, 2020

259. Total positivity in reductive groups, II, Bull. Inst. Math. Acad. Sinica (N.S.) 14(2019), 403-460.

260. On the totally positive grassmannian, arxiv:1905.09254.

261. The Grothendieck group of unipotent representations: a new basis, arxiv:1907.01401. to appear

## UCD MAT 280: Macdonald Polynomials and Crystal Bases - Mathematics

Today, November 30 th , is AMS Day! Join our celebration of AMS members and explore special offers on AMS publications, membership and more. Offers end 11:59pm EST.

ISSN 1088-6834(online) ISSN 0894-0347(print)

Schubert polynomials for the affine Grassmannian

Author: Thomas Lam

Journal: J. Amer. Math. Soc. **21** (2008), 259-281

MSC (2000): Primary 05E05 Secondary 14N15

DOI: https://doi.org/10.1090/S0894-0347-06-00553-4

Published electronically: October 18, 2006

MathSciNet review: 2350056

Full-text PDF Free Access

Abstract: Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the $k$-Schur functions in homology and affine Schur functions in cohomology. The results are obtained by connecting earlier combinatorial work of ours to certain subalgebras of Kostant and Kumar’s nilHecke ring and to work of Peterson on the homology of based loops on a compact group. As combinatorial corollaries, we settle a number of positivity conjectures concerning $k$-Schur functions, affine Stanley symmetric functions and cylindric Schur functions.

- Alberto Arabia,
*Cohomologie $T$-équivariante de la variété de drapeaux d’un groupe de Kac-Moody*, Bull. Soc. Math. France**117**(1989), no. 2, 129–165 (French, with English summary). MR**1015806** - I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand,
*Schubert cells, and the cohomology of the spaces $G/P$*, Uspehi Mat. Nauk**28**(1973), no. 3(171), 3–26 (Russian). MR**0429933** - Roman Bezrukavnikov, Michael Finkelberg, and Ivan Mirković,
*Equivariant homology and $K$-theory of affine Grassmannians and Toda lattices*, Compos. Math.**141**(2005), no. 3, 746–768. MR**2135527**, DOI https://doi.org/10.1112/S0010437X04001228 - Raoul Bott,
*The space of loops on a Lie group*, Michigan Math. J.**5**(1958), 35–61. MR**102803** - Paul Edelman and Curtis Greene,
*Balanced tableaux*, Adv. in Math.**63**(1987), no. 1, 42–99. MR**871081**, DOI https://doi.org/10.1016/0001-8708%2887%2990063-6 - Sergey Fomin and Curtis Greene,
*Noncommutative Schur functions and their applications*, Discrete Math.**193**(1998), no. 1-3, 179–200. Selected papers in honor of Adriano Garsia (Taormina, 1994). MR**1661368**, DOI https://doi.org/10.1016/S0012-365X%2898%2900140-X - Sergey Fomin and Richard P. Stanley,
*Schubert polynomials and the nil-Coxeter algebra*, Adv. Math.**103**(1994), no. 2, 196–207. MR**1265793**, DOI https://doi.org/10.1006/aima.1994.1009 - D. Gaitsgory,
*Construction of central elements in the affine Hecke algebra via nearby cycles*, Invent. Math.**144**(2001), no. 2, 253–280. MR**1826370**, DOI https://doi.org/10.1007/s002220100122 - Howard Garland and M. S. Raghunathan,
*A Bruhat decomposition for the loop space of a compact group: a new approach to results of Bott*, Proc. Nat. Acad. Sci. U.S.A.**72**(1975), no. 12, 4716–4717. MR**417333**, DOI https://doi.org/10.1073/pnas.72.12.4716 Gin V. Ginzburg: Perverse sheaves on a loop group and Langlands’ duality, preprint math.AG/9511007 . - William Graham,
*Positivity in equivariant Schubert calculus*, Duke Math. J.**109**(2001), no. 3, 599–614. MR**1853356**, DOI https://doi.org/10.1215/S0012-7094-01-10935-6 - Mark Haiman,
*Hilbert schemes, polygraphs and the Macdonald positivity conjecture*, J. Amer. Math. Soc.**14**(2001), no. 4, 941–1006. MR**1839919**, DOI https://doi.org/10.1090/S0894-0347-01-00373-3 - James E. Humphreys,
*Reflection groups and Coxeter groups*, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR**1066460** - Bertram Kostant and Shrawan Kumar,
*The nil Hecke ring and cohomology of $G/P$ for a Kac-Moody group $G$*, Proc. Nat. Acad. Sci. U.S.A.**83**(1986), no. 6, 1543–1545. MR**831908**, DOI https://doi.org/10.1073/pnas.83.6.1543 - Bertram Kostant and Shrawan Kumar,
*$T$-equivariant $K$-theory of generalized flag varieties*, Proc. Nat. Acad. Sci. U.S.A.**84**(1987), no. 13, 4351–4354. MR**894705**, DOI https://doi.org/10.1073/pnas.84.13.4351 - Shrawan Kumar,
*Kac-Moody groups, their flag varieties and representation theory*, Progress in Mathematics, vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. MR**1923198**Lam T. Lam: Affine Stanley symmetric functions,*Amer. J. Math.*, to appear math.CO/0501335 . LamAS T. Lam: Schubert polynomials for the affine Grassmannian (extended abstract),*Proc. FPSAC*, 2006, San Diego. LLMS T. Lam, L. Lapointe, J. Morse, and M. Shimozono: Affine insertion and Pieri rules for the affine Grassmannian, preprint, 2006 arXiv:math.CO/0609110. LamS T. Lam and M. Shimozono: A Little bijection for affine Stanley symmetric functions, preprint, 2006 math.CO/0601483 . - L. Lapointe, A. Lascoux, and J. Morse,
*Tableau atoms and a new Macdonald positivity conjecture*, Duke Math. J.**116**(2003), no. 1, 103–146. MR**1950481**, DOI https://doi.org/10.1215/S0012-7094-03-11614-2 - L. Lapointe and J. Morse,
*Schur function analogs for a filtration of the symmetric function space*, J. Combin. Theory Ser. A**101**(2003), no. 2, 191–224. MR**1961543**, DOI https://doi.org/10.1016/S0097-3165%2802%2900012-2 LM04 L. Lapointe and J. Morse: A $k$-tableaux characterization of $k$-Schur functions, preprint, 2005 arXiv:math.CO/0505519. LM05 L. Lapointe and J. Morse: Quantum cohomology and the $k$-Schur basis,*Tran. Amer. Math. Soc.*, to appear arXiv:math.CO/0501529. - Alain Lascoux and Marcel-Paul Schützenberger,
*Polynômes de Schubert*, C. R. Acad. Sci. Paris Sér. I Math.**294**(1982), no. 13, 447–450 (French, with English summary). MR**660739** - Alain Lascoux and Marcel-Paul Schützenberger,
*Schubert polynomials and the Littlewood-Richardson rule*, Lett. Math. Phys.**10**(1985), no. 2-3, 111–124. MR**815233**, DOI https://doi.org/10.1007/BF00398147 - I. G. Macdonald,
*Symmetric functions and Hall polynomials*, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. MR**553598**Pet D. Peterson: Lecture notes at MIT, 1997. - Alexander Postnikov,
*Affine approach to quantum Schubert calculus*, Duke Math. J.**128**(2005), no. 3, 473–509. MR**2145741**, DOI https://doi.org/10.1215/S0012-7094-04-12832-5 - Andrew Pressley and Graeme Segal,
*Loop groups*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR**900587**

- Ara A. Arabia: $T$-équivariante de la variété de drapeaux d’un groupe de Kac-Moody,

*Bull. Soc. Math. France*

**117**(1989), no. 2, 129–165. BGG I.N. Bernstein, I.M. Gelfand, and S.I. Gelfand: Schubert cells and cohomology of the spaces $G/P$,

*Russ. Math. Surv.*

**28**(1973), 1–26. BFM R. Bezrukavnikov, M. Finkelberg, and I. Mirković: Equivariant homology and $K$-theory of affine Grassmannians and Toda lattices,

*Compos. Math.*

**141**(2005), no. 3, 746–768. Bot R. Bott: The space of loops on a Lie group,

*Michigan Math. J.*

**5**(1958), 35–61. EG P. Edelman and C. Greene: Balanced tableaux,

*Adv. Math.*

**63**1 (1987), 42–99. FG S. Fomin and C. Greene: Noncommutative Schur functions and their applications,

*Discrete Mathematics*

**193**(1998), 179-200. FS S. Fomin and R. Stanley: Schubert polynomials and the nilCoxeter algebra,

*Adv. Math.*

**103**(1994), 196–207. Gai D. Gaitsgory: Construction of central elements in the affine Hecke algebra via nearby cycles,

*Invent. Math.*

**144**(2001), 253–280. GR H. Garland and M. S. Raghunathan: A Bruhat decomposition for the loop space of a compact group: a new approach to results of Bott,

*Proc. Nat. Acad. Sci. U.S.A.*

**72**(1975), no. 12, 4716–4717. Gin V. Ginzburg: Perverse sheaves on a loop group and Langlands’ duality, preprint math.AG/9511007 . Gra W. Graham: Positivity in equivariant Schubert calculus,

*Duke Math. J.*

**109**, no. 3 (2001), 599–614. Hai M. Haiman: Hilbert schemes, polygraphs, and the Macdonald positivity conjecture,

*J. Amer. Math. Soc.*

**14**(2001), 941–1006. Hum J. Humphreys: Reflection groups and Coxeter groups,

*Cambridge Studies in Advanced Mathematics*

**29**, Cambridge University Press, Cambridge, 1990. KK B. Kostant and S. Kumar: The nil Hecke ring and the cohomology of $G/P$ for a Kac-Moody group $G$,

*Adv. in Math.*

**62**(1986), 187–237. KK2 B. Kostant and S. Kumar: $T$-equivariant $K$-theory of generalized flag varieties,

*Proc. Natl. Acad. Sci. USA*

**84**(1987), 4351–4354. Kum S. Kumar: Kac-Moody groups, their flag varieties and representation theory,

*Progress in Mathematics*

**204**, Birkhäuser Boston, Inc., Boston, MA, 2002. Lam T. Lam: Affine Stanley symmetric functions,

*Amer. J. Math.*, to appear math.CO/0501335 . LamAS T. Lam: Schubert polynomials for the affine Grassmannian (extended abstract),

*Proc. FPSAC*, 2006, San Diego. LLMS T. Lam, L. Lapointe, J. Morse, and M. Shimozono: Affine insertion and Pieri rules for the affine Grassmannian, preprint, 2006 arXiv:math.CO/0609110. LamS T. Lam and M. Shimozono: A Little bijection for affine Stanley symmetric functions, preprint, 2006 math.CO/0601483 . LLM L. Lapointe, A. Lascoux, and J. Morse: Tableau atoms and a new Macdonald positivity conjecture,

*Duke Math. J.*

**116**(1) (2003), 103–146. LM L. Lapointe and J. Morse: Schur function analogs for a filtration of the symmetric function space,

*J. Combin. Theory Ser. A*

**101**(2) (2003), 191-224. LM04 L. Lapointe and J. Morse: A $k$-tableaux characterization of $k$-Schur functions, preprint, 2005 arXiv:math.CO/0505519. LM05 L. Lapointe and J. Morse: Quantum cohomology and the $k$-Schur basis,

*Tran. Amer. Math. Soc.*, to appear arXiv:math.CO/0501529. LS A. Lascoux and M. Schützenberger: Polynômes de Schubert,

*C.R. Acad. Sci. Paris*,

**294**(1982), 447–450. LS85 A. Lascoux and M. Schützenberger: Schubert polynomials and the Littlewood-Richardson rule,

*Lett. Math. Phys.*

**10**(2-3) (1985), 111–124. Mac I. G. Macdonald:

*Symmmetric Functions and Hall Polynomials,*Oxford, 1970. Pet D. Peterson: Lecture notes at MIT, 1997. Pos A. Postnikov: Affine approach to quantum Schubert calculus,

*Duke Math. J.*

**128**(3) (2005), 473–509. PS A. Pressley and G. Segal: Loop groups, Clarendon Press, Oxford, 1986.

Retrieve articles in *Journal of the American Mathematical Society* with MSC (2000): 05E05, 14N15

Retrieve articles in all journals with MSC (2000): 05E05, 14N15

**Thomas Lam**

Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

MR Author ID: 679285

ORCID: 0000-0003-2346-7685

Email: [email protected]

Keywords: Schubert polynomials, symmetric functions, Schubert calculus, affine Grassmannian

Received by editor(s): April 7, 2006

Published electronically: October 18, 2006

Article copyright: © Copyright 2006 American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.

## Orthogonal Polynomials of Several Variables

##### This book has been cited by the following publications. This list is generated based on data provided by CrossRef.

- Publisher: Cambridge University Press
- Online publication date: August 2014
- Print publication year: 2014
- Online ISBN: 9781107786134
- DOI: https://doi.org/10.1017/CBO9781107786134

- Subjects: Mathematics (general), Mathematics, Real and Complex Analysis
- Series: Encyclopedia of Mathematics and its Applications (155)

Email your librarian or administrator to recommend adding this book to your organisation's collection.

### Book description

Serving both as an introduction to the subject and as a reference, this book presents the theory in elegant form and with modern concepts and notation. It covers the general theory and emphasizes the classical types of orthogonal polynomials whose weight functions are supported on standard domains. The approach is a blend of classical analysis and symmetry group theoretic methods. Finite reflection groups are used to motivate and classify symmetries of weight functions and the associated polynomials. This revised edition has been updated throughout to reflect recent developments in the field. It contains 25% new material, including two brand new chapters on orthogonal polynomials in two variables, which will be especially useful for applications, and orthogonal polynomials on the unit sphere. The most modern and complete treatment of the subject available, it will be useful to a wide audience of mathematicians and applied scientists, including physicists, chemists and engineers.

### Reviews

Review of the first edition:‘This book is the first modern treatment of orthogonal polynomials of several real variables. It presents not only a general theory, but also detailed results of recent research on generalizations of various classical cases.'

Source: Mathematical Reviews

Review of the first edition:‘This book is very impressive and shows the richness of the theory.'

Vilmos Totik Source: Acta Scientiarum Mathematicarum

‘This is a valuable book for anyone with an interest in special functions of several variables.'

## UCD MAT 280: Macdonald Polynomials and Crystal Bases - Mathematics

You can also see the latest edition or all old editions. To search old editions, use the box above and add "week" to your search terms.

Many of the papers I review are available on the arXiv. If you click on the number of one of these papers, such as arXiv:hep-th/9301028, a magic carpet will carry you to a place where you can read an abstract of the paper, and download it if you like. Click here for more information on how to get papers electronically.

### Week1

On formulations and solutions of simplex equations, by J. Scott Carter and Masahico Saito, Intern. J. of Mod. Phys. A., 11 (1996) 4453-4463.

A diagrammatic theory of knotted surfaces, by J. Scott Carter and Masahico Saito, in Quantum Topology, eds. Randy Baadhio and Louis Kauffman, World Science Publishing, Singapore, 1993, 91-115.

Reidemeister moves for surface isotopies and their interpretations as moves to movies, by J. Scott Carter and Masahico Saito, Journal of Knot Theory and its Ramifications 2 (1993), 251-284.

### Week2

A Categorical construction of 4d topological quantum field theories, by Louis Crane and David Yetter, preprint available as arXiv:hep-th/9301062.

Hopf Categories and their representations, Louis Crane and Igor Frenkel, draft version.

Categorification and the construction of topological quantum field theory, Louis Crane and Igor Frenkel, draft version.

### Week3

### Week4

### Week5

### Week6

### Week7

### Week8

Map coloring, 1-deformed spin networks, and Turaev-Viro invariants for 3-manifolds, by Louis Kauffman, Int. Jour. of Mod. Phys. B, 6 (1992) 1765 - 1794.

An algebraic approach to the planar colouring problem, by Louis Kauffman and H. Saleur, in Comm. Math. Phys. 152 (1993), 565-590.

Spin networks, topology and discrete physics, by Louis Kauffman, University of Illinois at Chicago preprint.

Vassiliev invariants and the Jones polynomial, by Louis Kauffman, University of Illinois at Chicago preprint.

Gauss codes and quantum groups, by Louis Kauffman, University of Illinois at Chicago preprint.

Fermions and link invariants, by Louis Kauffman and H. Saleur, Yale University preprint YCTP-P21-91, July 5, 1991.

State models for link polynomials, by Louis Kauffman, L'Enseignement Mathematique, 36 (1990), 1 - 37.

The Conway polynomial in R^3 and in thickened surfaces: a new determinant formulation, by F. Jaeger, Louis Kauffman and H. Saleur, preprint.

### Week9

Reshetikhin-Turaev and Crane-Kohno-Kontsevich 3-manifold invariants coincide, by Sergey Piunikhin, preprint, 1992. (Piunikhin is at [email protected])

A link calculus for 4-manifolds, by E. Cesar de Sa, in Topology of Low-Dimensional Manifolds, Proc. Second Sussex Conf., Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 16-30,

A note on 4-dimensional handlebodies, by F. Laudenbach and V. Poenaru, Bull. Math. Soc. France 100 (1972), pp. 337-344,

Minisuperspaces: observables and quantization, Abhay Ashtekar, Ranjeet S. Tate and Claes Uggla Syracuse University preprint SU-GP-92/2-6, 34 pages, available as arXiv:gr-qc/9302027

### Week10

Turaev-Viro and Kauffman-Lins invariants for 3-manifolds coincide, by Sergey Piunikhin, Journal of Knot Theory and its Ramifications, 1 (1992) 105 - 135.

Different presentations of 3-manifold invariants arising in rational conformal field theory, by Sergey Piunikhin, preprint.

Weights of Feynman diagrams, link polynomials and Vassiliev knot invariants, by Sergey Piunikhin, preprint.

Reshetikhin-Turaev and Crane-Kohno-Kontsevich 3-manifold invariants coincide, by Sergey Piunikhin, preprint.

### Week11

### Week12

### Week13

### Week14

### Week15

### Week16

### Week17

### Week18

### Week19

### Week20

### Week21

### Week22

Map coloring, 1-deformed spin networks, and Turaev-Viro invariants for 3-manifolds, by Louis Kauffman, Int. Jour. of Mod. Phys. B, 6 (1992) 1765 - 1794.

An algebraic approach to the planar colouring problem, by Louis Kauffman and H. Saleur, Yale University preprint YCTP-P27-91, November 8, 1991.

### Week23

The loop formulation of gauge theory and gravity, by Renate Loll

Representation theory of analytic holonomy C* algebras, by Abhay Ashtekar and Jerzy Lewandowski (currently available as arXiv:gr-qc/9311010)

The Gauss linking number in quantum gravity, by Rodolfo Gambini and Jorge Pullin (currently available as arXiv:gr-qc/9310025)

Vassiliev invariants and the loop states in quantum gravity, by Louis H. Kauffman (soon to be on gr-qc)

Geometric structures and loop variables in (2+1)-Dimensional gravity, by Steven Carlip (currently available as arXiv:gr-qc/9309020)

From Chern-Simons to WZW via path integrals, by Dana S. Fine

Topological field theory as the key to quantum gravity, by Louis Crane (currently available as arXiv:hep-th/9308126)

Strings, loops, knots and gauge gields, by John Baez (currently available as arXiv:hep-th/9309067 and also at string.tex).

BF Theories and 2-knots, by Paolo Cotta-Ramusino and Maurizio Martellini

Knotted surfaces, braid movies, and beyond, by J. Scott Carter and Masahico Saito

### Week24

### Week25

### Week26

### Week27

Loss of quantum coherence for a damped oscillator, by W. G. Unruh, the volume above.

The problem of time in canonical quantization of relativistic systems, by Karel V. Kuchar, the volume above.

Time and prediction in quantum cosmology, by James B. Hartle, the volume above.

Space and time in the quantum universe, by Lee Smolin, the volume above.

Loop representation in quantum gravity, by Carlo Rovelli, the volume above.

Nonperturbative quantum gravity via the loop representation, by Lee Smolin, the volume above.

### Week28

### Week29

A presentation for Manin and Schechtman's higher braid groups, by R. J. Lawrence, available as MSRI preprint 04129-91.

Triangulations, categories and extended topological field theories, by R. J. Lawrence, in Quantum Topology, eds L. Kauffman and R. Baadhio, World Scientific, Singapore, 1993.

Algebras and triangle relations, by R. J. Lawrence, Harvard U. preprint.

Coherence for the tensor product of 2-categories, and braid groups, in Algebras, Topology, and Category Theory, eds. A. Heller and M. Tierney, Academic Press, New York, 1976, pp. 63-76.

### Week30

Kepler's Physical Astronomy, by Bruce Stephenson, Princeton U. Press, 218 pages, paperback available June 1994. ISBN 0-691-03652-7 ($14.95).

Fermions in quantum gravity, by H. A. Morales-Tecotl and C. Rovelli, 37 pages, preprint available as arXiv:gr-qc/9401011.

### Week31

Black hole entropy in canonical quantum gravity and superstring theory, by L. Susskind and J. Uglum, 29 pages, available as arXiv:hep-th/9401070.

### Week32

### Week33

### Week34

Quantum theory, the Church--Turing principle and the universal quantum computer, by D. Deutsch, Proc. R. Soc. Lond., Vol. A400, pp. 96--117 (1985).

Quantum computational networks, by D. Deutsch, Proc. R. Soc. Lond., Vol. A425, pp. 73--90 (1989).

Rapid solution of problems by quantum computation, by D. Deutsch and R. Jozsa, Proc. R. Soc. Lond., Vol. A439, pp. 553--558 (1992).

Quantum complexity theory, E. Bernstein and U. Vazirani, Proc. 25th ACM Symp. on Theory of Computation, pp. 11--20 (1993).

The quantum challenge to structural complexity theory, A. Berthiaume and G. Brassard, Proc. 7th IEEE Conference on Structure in Complexity Theory (1992).

Quantum circuit complexity, by A. Yao, Proc. 34th IEEE Symp. on Foundations of Computer Science, 1993.

### Week35

### Week36

### Week37

### Week38

A definition of #(M,H) in the non-involutory case, by Greg Kuperberg, unpublished.

Invariants of piecewise-linear 3-manifolds, by John W. Barrett and Bruce W. Westbury, Trans. Amer. Math. Soc. 348 (1996), 3997-4022, preprint available as arXiv:hep-th/9311155.

The equality of 3-manifold invariants, by John W. Barrett and Bruce W. Westbury, preprint available as arXiv:hep-th/9406019.

Topological measure and graph-differential geometry on the quotient space of connections, Jerzy Lewandowski, 3 pp., Proceedings of ``Journees Relativistes 1993'', 3 pages available as arXiv:gr-qc/9406025.

Integration on the space of connections modulo gauge transformations, Abhay Ashtekar, Donald Marolf, Jose Mourao, 18 pages, preprint available as arXiv:gr-qc/9403042.

New loop representations for 2+1 gravity, by A. Ashtekar and R. Loll, 28 pages, preprint available as arXiv:gr-qc/9405031.

Independent loop invariants for 2+1 gravity, by R. Loll, 2 figures, arXiv:gr-qc/9408007.

Generalized coordinates on the phase space of Yang-Mills theory, by R. Loll, J.M. Mour

ao and J.N. Tavares, 11 pages, preprint available as arXiv:gr-qc/9404060.

The extended loop representation of quantum gravity, C. Di Bartolo, R. Gambini and J. Griego, 27 pages available as arXiv:gr-qc/9406039.

The constraint algebra of quantum gravity in the loop representation, by Rodolfo Gambini, Alcides Garat and Jorge Pullin, 18 pages in Revtex, available as arXiv:gr-qc/9404059.

### Week39

Invariant functions on Lie groups and Hamiltonian flows of surface group representations, by W. Goldman, Invent. Math. 83 (1986), 263-302.

Topological components of spaces of representations, by W. Goldman, Invent. Math. 93 (1988), 557-607.

### Week40

### Week41

### Week42

Gauge fields as rings of glue, A. Polyakov, Nucl. Phys. B164 (1979) 171-188.

The quantum dual string wave functional in Yang-Mills theories, by J. Gervais and A. Neveu, Phys. Lett. B80 (1979), 255-258.

The interaction among dual strings as a manifestation of the gauge group, by F. Gliozzi and M. Virasoro, Nucl. Phys. B164 (1980), 141-151.

Loop-space representation and the large-N behavior of the one-plaquette Kogut-Susskind Hamiltonian, A. Jevicki, Phys. Rev. D22 (1980), 467-471.

Quantum chromodynamics as dynamics of loops, by Y. Makeenko and A. Migdal, Nucl. Phys. B188 (1981) 269-316.

Loop dynamics: asymptotic freedom and quark confinement, by Y. Makeenko and A. Migdal, Sov. J. Nucl. Phys. 33 (1981) 882-893.

String Fields, Conformal Fields, and Topology, by Michio Kaku, New York, Springer-Verlag, 1991.

Background independent algebraic structures in closed string field theory, by Ashoke Sen and Barton Zwiebach, phyzzx.tex, MIT-CTP-2346, available as arXiv:hep-th/9408053.

U(N) Gauge Theory and lattice strings, by Ivan K. Kostov, 26 pages, 8 figures not included, available by mail upon request, T93-079 (talk at the Workshop on string theory, gauge theory and quantum gravity, 28-29 April 1993, Trieste, Italy), available as arXiv:hep-th/9308158.

### Week43

Quantum geometrodynamics, by A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao and T. Thiemann, in progress, to appear on gr/qc.

On the constraints of quantum gravity in the loop representation, Bernd Bruegmann and Jorge Pullin, Nucl. Phys. B390 (1993), 399-438.

On the constraints of quantum general relativity in the loop representation, Bernd Bruegmann, Ph.D. Thesis, Syracuse University (May 1993).

### Week44

Polynomial invariants for smooth four-manifolds, by S. K. Donaldson, Topology 29 (1990), 257-315.

"Instantons and Four-Manifolds," by Daniel S. Freed and Karen K. Uhlenbeck, Springer-Verlag, New York (1984).

"Differential Topology and Quantum Field Theory," by Charles Nash, Academic Press, London, 1991.

### Week45

Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, by Nathan Seiberg and Edward Witten, 45 pages, available as arXiv:hep-th/9407087.

Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, by Nathan Seiberg and Edward Witten, 89 pages, available as arXiv:hep-th/9408099.

### Week46

Goodbye, Gutenberg, by Jacques Leslie, WiReD 2.10, Oct. 1994, available via WWW as http://www.hotwired.com/Lib/Wired/2.10/departments /electrosphere/ejournals.html

The genus of embedded surfaces in the projective plane, by P. B. Kronheimer and T. S. Mrowka, preprint number #19941128001, available from the AMS preprint server under subject 57 in the Mathematical Reviews Subject Classification Scheme.

Coherent state transforms for spaces of connections, by Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao and Thomas Thiemann, Jour. Funct. Analysis 135 (1996), 519-551, preprint available as arXiv:gr-qc/9412014 (discussed in "week43")

Reminiscences about many pitfalls and some successes of QFT within the last three decades, by B. Schroer, 52 pages, 'shar'-shell-archiv, consisting of 5 files, available as arXiv:hep-th/9410085.

My encounters - as a physicist - with mathematics, R. Jackiw, 13 pages, preprint available as arXiv:hep-th/9410151.

### Week47

### Week48

### Week49

### Week50

The semiclassical limit of the two-dimensional quantum Yang-Mills model, same authors, Jour. Math. Phys. 35 (1994), 5354-5363.

### Week51

### Week52

Alberto Cattaneo, Paolo Cotta-Ramusino, Juerg Froehlich, and Maurizio Martellini, Topological BF theories in 3 and 4 dimensions, preprint available as arXiv:hep-th/9505027.

### Week53

G. Kelly, Structures defined by finite limits in the enriched context I, Cahiers de Top. et. Geom. Diff. 23 (1982), 3-41.

Michael Makkai and Robert Pare, Accessible categories: the foundations of categorical model theory, in Contemp. Math. 104 (1989).

### Week54

TQFTs from homotopy 2-types, Journal of Knot Theory and its Ramifications 2 (1993), 113-123.

Refined state-sum invariants of 3- and 4-manifolds, preprint.

Skeins and mapping class groups, Math. Proc. Camb. Phil. Soc. 115 (1994), 53-77.

G. Masbaum and Justin Roberts, On central extensions of mapping class groups, Mathematica Gottingensis, Schriftenreihe des Sonderforschungsbereichs Geometrie und Analysis, Heft 42 (1993).

### Week55

### Week56

### Week57

L. Susskind, Strings, black holes and Lorentz contractions, preprint available as arXiv:hep-th/9308139.

### Week58

### Week59

Octonion X-product and E8 lattices, preprint available as arXiv:hep-th/9411063.

Octonions: E8 lattice to Lambda_<16>, preprint available as arXiv:hep-th/9501007.

Octonions: invariant representation of the Leech lattice, preprint available as arXiv:hep-th/9504040.

Octonions: invariant Leech lattice exposed, preprint available as arXiv:hep-th/9506080.

### Week60

L. Rozansky, The trivial connection contribution to Witten's invariant and finite type invariants of rational homology spheres, preprint available as arXiv:q-alg/9505015.

Stavros Garoufalidis, On finite type 3-manifold invariants I, MIT preprint, 1995.

Stavros Garoufalidis and Jerome Levine, On finite type 3-manifold invariants II, MIT preprint, June 1995. (Garoufalidis is at [email protected], and Levine is at [email protected])

Ruth J. Lawrence, Asymptotic expansions of Witten-Reshetikhin-Turaev invariants for some simple 3-manifolds, to appear in Jour. Math. Physics.

### Week61

### Week62

### Week63

### Week64

### Week65

John McKay, Representations and Coxeter Graphs, in "The Geometric Vein" Coxeter Festschrift (1982), Springer-Verlag, Berlin, pages 549-.

John McKay, A rapid introduction to ADE theory, http://math.ucr.edu/home/baez/ADE.html

### Week66

Richard E. Borcherds, Monstrous Moonshine and monstrous Lie-superalgebras, Invent. Math. 109 (1992), 405-444.

### Week67

### Week68

J. Sexton, A. Vaccarino, D. Weingarten, Numerical evidence for the observation of a scalar glueball, available as arXiv:hep-lat/9510022.

Quantization of Poisson algebraic groups and Poisson homogeneous spaces, preprint available in AMSTeX form as arXiv:q-alg/9510020.

Claudio Teitelboim, Statistical thermodynamics of a black hole in terms of surface fields, preprint available as arXiv:hep-th/9510180.

The Kauffman bracket and the Jones polynomial in quantum gravity, preprint available as arXiv:gr-qc/9510050.

### Week69

### Week70

Graeme Mitchison and Richard Jozsa, Counterfactual quantum computation, Proc. Roy. Soc. Lond. A457 (2001) 1175-1194. Also available as quant-ph/9907007.

Jean-Yves Girard, Y. Lafont and P. Taylor, Proofs and Types, Cambridge Tracts in Theoretical Computer Science 7, Cambridge U. Press, 1989. Also available at http://www.cs.man.ac.uk/

Eric Goubault and Thomas Jensen, Homology of higher-dimensional automata, in Proc. CONCUR '92 (New York), Lecture Notes in Computer Science 630, Springer, 1992. Also available at http://www.di.ens.fr/%7Egoubault/GOUBAULTpapers.html

Craig C. Squier, A finiteness condition for rewriting systems, revision by F. Otto and Y. Kobayashi, to appear in Theoretical Computer Science.

Craig C. Squier and F. Otto, The word problem for finitely presented monoids and finite canonical rewriting systems, in J. P. Jouannuad (ed.), Rewriting Techniques and Applications, Lecture Notes in Computer Science 256 (1987), 74-82.

Yves Lafont and Alain Proute, Church-Rosser property and homology of monoids, Mathematical Structures in Computer Science 1 (1991), 297-326. Also available at http://iml.univ-mrs.fr/

Yves Lafont, A new finiteness condition for monoids presented by complete rewriting systems (after Craig C. Squier), Journal of Pure and Applied Algebra 98 (1995), 229-244. Also available at http://iml.univ-mrs.fr/

### Week71

George Thompson, New results in topological field theory and abelian gauge theory, 64 pages, preprint available as arXiv:hep-th/9511038.

Thomas Thiemann, Reality conditions inducing transforms for quantum gauge field theory and quantum gravity, preprint available as arXiv:gr-qc/9511057.

Abhay Ashtekar, A generalized Wick transform for gravity, preprint available as arXiv:gr-qc/9511083.

Renate Loll, Making quantum gravity calculable, preprint available as arXiv:gr-qc/9511080.

Rodolfo Gambini and Jorge Pullin, A rigorous solution of the quantum Einstein equations, preprint avilable in RevTex form as arXiv:gr-qc/9511042, four figures included with epsf.

Lev Rozansky, On finite type invariants of links and rational homology spheres derived from the Jones polynomial and Witten- Reshetikhin-Turaev invariant, preprint available as arXiv:q-alg/9511025.

Scott Axelrod, Overview and warmup example for perturbation theory with instantons, preprint available as arXiv:hep-th/9511196.

Alan Carey, M. K. Murray and B. L. Wang, Higher bundle gerbes and cohomology classes in gauge theories, preprint available as arXiv:hep-th/9511169

Jean-Luc Brylinski and D. A. McLaughlin, The geometry of degree-four characteristic classes and of line bundles on loop spaces I, Duke Math. Jour. 75 (1994), 603-638.

Jean-Luc Brylinski and D. A. McLaughlin, Cech cocyles for characteristic classes.

### Week72

Roumen Borissov, Seth Major and Lee Smolin, The geometry of quantum spin networks, preprint available as arXiv:gr-qc/9512043, 35 Postscript figures, uses epsfig.sty.

Bernd Bruegmann, On the constraint algebra of quantum gravity in the loop representation, preprint available as arXiv:gr-qc/9512036.

Kiyoshi Ezawa, Nonperturbative solutions for canonical quantum gravity: an overview, preprint available as arXiv:gr-qc/9601050

Kiyoshi Ezawa, A semiclassical interpretation of the topological solutions for canonical quantum gravity, preprint available as arXiv:gr-qc/9512017.

Jorge Griego, Extended knots and the space of states of quantum gravity, preprint available as arXiv:gr-qc/9601007.

Seth Major and Lee Smolin, Quantum deformation of quantum gravity, preprint available as arXiv:gr-qc/9512020.

### Week73

D. K. Kondepudi and D. K. Nelson, Weak neutral currents and the origins of molecular chirality, Nature 314, pp. 438-441.

### Week74

Jerzy Lewandowski, Volume and Quantizations, preprint available as arXiv:gr-qc/9602035.

H. Weyl, in *Symmetry* (Princeton University Press, New Jersey, 1952). There is an extended edition (contains also articles by other authors), ed. by B.A. Rosenfeld (Nauka, Moscow, 1968) **(in Russian)**

L. Pauling, R. Hayward, *The Architecture of Molecules* (W. H. Freeman and Company, San Francisco, 1964)

A.V. Shubnikov, N.V. Belov, W.T.E. Holser, *Color Symmetry* (Pergamon Press, Oxford, 1964)

A.V. Shubnikov, *U Istokov Kristallografii (At the Dawn of Crystallography)* (Nauka, Moscow, 1971). (in Russian)

A.V. Shubnikov, V.A. Koptsik, *Symmetry in Science and Art* (Plenum Press, Berlin, 1974). (translated from Russian)

YuA Urmantsev, *Simmetriya Prirody i Priroda Simmetrii (Symmetry of Nature and Nature of Symmetry)* (Mysl’, Moscow, 1974). (in Russian)

M. Senechal, G. Fleck, *Patterns of Symmetry* (University of Massachusetts Press, Amherst, 1977)

J.F. Sadoc, R. Mosseri, *Geometrical Frustration* (Cambridge University Press, Cambridge, 1999)

I. Hargittai, T.C. Laurent, *Symmetry 2000*, vol. 1 (Portland Press Ltd., London, 2002)

I. Hargittai, T.C. Laurent, *Symmetry 2000*, vol. 2 (Portland Press Ltd., London, 2002)

L. Michel , E. Brezin (eds.), Symmetry, invariants, topology. Phys. Rep. **341**, 1–368 (2001)

K. Mainzer, in *Symmetry and Complexity. The Spirit and Beauty of Nonlinear Science, World Scientific Series on Nonlinear Science, Series A*, vol. 51, ed. by L.O. Chua (World Scientific, Singapore, 2005)

E.A. Lord, A.L. Mackay, S. Ranganathan, *New Geometries for New Materials* (Cambridge University Press, Cambridge, 2006)

V.V. Iliev, *Isomerism as Intrinsic Symmetry of Molecules*, Mathematical Chemistry Monographs, No 5 (University of Kragujevac and Faculty of Science, Kragujevac, 2008)

D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. **53**, 1951–1954 (1984)

L.S. Levitov, Local rules for quasicrystals. Commun. Math. Phys. **119**, 627–666 (1988)

E.A. Lord, Quasicrystals and Penrose patterns. Curr. Sci. **61**(5), 313–319 (1991)

D. Baraches, S. De Bievre, J.-P. Gazeau, Affine symmetry semi-groups for quasi-crystals. Europhys. Lett. **25**(6), 435–440 (1994)

P.J. Steinhardt, New perspectives on forbidden symmetries, quasicrystals, and Penrose tilings. Proc. Natl. Acad. Sci. USA **93**, 14267–14270 (1996)

R. Lifshitz, Theory of color symmetry for periodic and quasiperiodic crystals. Rev. Mod. Phys. **69**(4), 1181–1216 (1997)

R. Lifshitz, Lattice color groups of quasicrystals. *cond-math 9704105 v2 29 Jan 1998*

J.-P. Gazeau, J. Miȩkisz, A symmetry group of a Thue–Morse quasicrystal. arXiv:cond-mat/9904230v1 [cond-mat.stat-mech] (1999)

M.I. Samoïlovich, A.L. Talis, , M.I. Mironov, Quasicrystals with the infinite point group as a symmetry base of diamond-like structures. Dokl. Phys. **47**(6), 447–450 (2002). Translated from Doklady Akademii Nauk **384**(6), 760–763 (2002)

R.B. King, Regular polytopes, root lattices, and quasicrystals. Croat. Chem. Acta **77**(1–2), 133–140 (2004)

W. Steurer, Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals. Z. Kristallogr. **219**, 391–446 (2004)

V.A. Artamonov, Quasicrystals and their symmetries. J. Math. Sci. **139**(4), 6657–6662 (2006). Translated from Fundamentalnaya i Prikladnaya Matematika, **10**(3), 3–10 (2004)

E. Pelantová , Z. Masáková, Quasicrystals: algebraic, combinatorial and geometrical aspects. arXiv:math-ph/0603065v1 (2006)

W. Steurer, Reflections on symmetry and formation of axial quasicrystals. Z. Kristallogr. **221**, 402–411 (2006)

M. Senechal, What is.. a quasicrystal? Not. Am. Math. Soc. **53**(8), 886–887 (2006)

V.A. Artamonov, S. Sanchez, On symmetry groups of quasicrystals. Math. Notes **87**(3), 303–308 (2010). Original Russian text (copyright ) V.A. Artamonov, S. Sanchez, published in Matematicheskie Zametki 87(3), 323–329 (2010)

Y.K. Vekilov, M.A. Chernikov, Quasicrystals. Uspekhi Fiz. Nauk **180**(6), 561–586 (2010). (in Russian)

J. Mikhael, M. Schmiedeberg, S. Rausch, J. Roth, H. Stark, C. Bechinger, Proliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields. PNAS **107**(16), 7210–7218 (2010)

M. Baake, U. Grimm, On the notions of symmetry and aperiodicity for Delone sets. Symmetry **4**, 566–580 (2012)

V.V. Yudin, E.S. Startzev, The Fibonacci fractal is a new fractal type. Theor. Math. Phys. **173**(1), 1387–1402 (2012)

D. Ashkenazi, Z. Lotker, The quasicrystals discovery as a resonance of the non-Euclidean geometry revolution: historical and philosophical perspective. Philosophia **42**, 25–40 (2014). doi:10.1007/s11406-013-9504-8

V.R. Rosenfeld, A.A. Dobrynin, J.M. Oliva, J. Rué, Pentagonal chains and annuli as models for designing nanostructures from cages. J. Math. Chem. **54**(3), 765–776 (2016)

B.L. van der Waerden, J.J. Burckhardt, Farbgruppen. Z. Kristallogr. **115**, 231–234 (1961)

A.L. Loeb, *Color and Symmetry* (Wiley, New York, 1971)

M. Senechal, Point groups and color symmetry. Z. Kristallogr. **142**, 1–23 (1975)

C.H. MacGillavry, *Fantasy and Symmetry, the Periodic Drawings of MC Escher* (Harry N. Abrams, New York, 1976)

S.O. MacDonald, A.P. Street, On crystallographic color groups, in *Lecture Notes in Mathematics*, vol. 560 (Springer, Berlin, 1976), pp. 149–157

B. Grünbaum, G.C. Shephard, Perfect colorings of the transitive tilings and patterns in the plane. Discrete Math. **20**, 235–247 (1977)

S.O. MacDonald, A.P. Street, The analysis of color symmetry, in *Lecture Notes in Mathematics*, vol. 686 (Springer, Berlin, 1978), pp. 210–222

R.L.E. Schwarzenberger, *N-Dimensional Crystallography* (Pitman, San Francisco, 1980)

R.L. Roth, Color symmetry and group theory. Discrete Math. **38**, 273–296 (1982)

V.R. Rosenfeld, Color symmetry, semigroups, fractals. Croat. Chem. Acta **86**(4), 555–559 (2013)

M. Mucha, Hidden symmetries and Weyl’s recipe, in *Symmetry and Structural Properties of Condensed Matter*, ed. by W. Florek, T. Lulek, M. Mucha (World Scientific, Singapore, 1991), p. 19

B. Lulek, Impurities in the Heisenberg magnet and the general recipe of Weyl. Semin. Lothar. Combin. **26**, 7 (1991)

A. Kerber, *Applied Finite Group Actions* (Springer, Berlin, 1999)

V.R. Rosenfeld, On mathematical engineering and design of novel molecules for nanotechnological applications—review. Sci. Isr. Technol. Adv. **9**(1), 56–65 (2007)

V.R. Rosenfeld, Toward molecules with nonstandard symmetry, Ch 14, in *Diamond and Related Nanostructures*, ed. by M.V. Diudea, C.L. Nagy (Springer, Berlin, 2013), pp. 275–285

B.B. Mandelbrot, *The Fractal Geometry of Nature* (W. H. Freeman and Co., New York, 1982)

M.F. Barnsley, H. Rising, *Fractals Everywhere* (Academic Press, Boston, 1993)

J.F. Gouyet, *Physics and Fractal Structures* (B. Mandelbrot, Foreword and Springer, Masson and New York, 1996)

K. Falconer, *Techniques in Fractal Geometry* (Wiley, USA, 1997)

D.J. Klein, W.A. Seitz, J.E. Kilpatrick, Branched polymer models. J. Appl. Phys. **53**(10), 6599–6603 (1982)

D.J. Klein, W.A. Seitz, Self-similar self-avoiding structures: models for polymers. Proc. Natl. Acad. Sci. USA **80**, 3125–3128 (1983)

D.J. Klein, W.A. Seitz, Graphs, polymer models, excluded volume, and chemical reality, in *Topology and Graph Theory in Chemistry*, ed. by R.B. King (Elsevier, Amsterdam, 1983), pp. 430–445

D.J. Klein, Self-interacting self-avoiding walks on the Sierpinski gasket. J. Phys. Lett. **45**(6), L-241–L-247 (1984)

W.A. Seitz, D.J. Klein, G.E. Hite, Interacting dimers on a Sierpiński gasket. Discrete Appl. Math. **19**, 339–348 (1988)

D.J. Klein, T.P. Živković, A.T. Balaban, The fractal family of coro Nenes. MATCH Commun. Math. Comput. Chem. **29**, 107–130 (1993)

L. Bytautas, D.J. Klein, M. Randić, T. Pisanski, Foldedness in linear polymers: a difference between graphical and Euclidean distances. DIMACS Ser. Discrete Math. Theor. Comput. Sci. **51**, 39–61 (2000)

D.J. Klein, D. J, A.T. Balaban, Clarology for conjugated carbon nano-structures: molecules, polymers, graphene, defected graphene, fractal benzenoids, fullerenes, nano-tubes, nano-cones, nano-tori, etc. Open Org. Chem. J. (Suppl 1-M3) **5**, 27–61 (2011)

Y. Almirantis, A. Provata, An evolutionary model for the origin of non-random long-range order and fractality in the genome. BioEssays **23**, 647–656 (2001)

N.N. Oiwa, J.A. Glazier, The fractal structure of the mitochondrial genomes. Phys. A **311**, 221–230 (2002)

M.A. Moret, J.G. Miranda, E. Noqueira, Jr., M.C. Santana, G.F. Zebende, Self-similarity and protein chains. Phys. Rev. E Stat. Nonlinear Soft Matter. Phys. **71**(1 Pt 1), 012901 (2005) **(epub 2005 Jan 27)**

C. Cattani, Fractals and hidden symmetries in DNA. Math. Probl. Eng. **2010**, 507056 (2010). doi:10.1155/2010/507056

N. Todoroff, J. Kunze, H. Schreuder, K.-H. Baringhaus, G. Schneider, Fractal dimensions of macromolecular structures. Mol. Inf. **33**, 588–596 (2014)

R. Hancock, Structures and functions in the crowded nucleus: new biophysical insights. Front. Phys. **53**, 1–7 (2014)

V.R. Rosenfeld, The fractal nature of folds and the Walsh copolymers. J. Math. Chem. **54**(2), 559–571 (2016)

C. Stover, E.W. Weisstein, “Groupoid” from MathWorld—a wolfram web resource. http://mathworld.wolfram.com/Groupoid.html

M. Lothaire, *Combinatorics on Words* (Addison-Wesley, USA, 1983)

I. Dolinka, J. East, Twisted Braurer monoids. arXiv:1510.08666v1 [math.GR] (2015)

F. Frucht, Graphs of degree three with a given abstract group. Can. J. Math. **1**, 365–378 (1949)

G. Sabidussi, Graphs with given group and given graph-theoretical properties. Can. J. Math. **9**, 515–525 (1957)

F. Harary, *Graph Theory* (Addison-Wesley, Reading, 1969)

D.M. Cvetković, M. Doob, H. Sachs, *Spectra of Graphs-Theory and Application* (VEB Deutscher Verlag der Wissenschaften, Berlin, 1980)

W. Burnside, *Theory of Groups of Finite Order* (Cambridge, 1911). Project Gutenberg-tm electronic works, EBook #40395, Release date: August 2, 2012. www.gutenberg.org/ebooks/

H.S.M. Coxeter, W.O.J. Moser, *Generators and Relations for Discrete Groups* (Springer, Berlin, 1957)

M. Hall Jr., *The Theory of Groups* (Macmillan, New York, 1959)

A.G. Kurosh, *The Theory of Groups*, vol. 1 (Chelsea, New York, 1960). (translated from Russian)

A.G. Kurosh, *The Theory of Groups*, vol. 2 (Chelsea, New York, 1960). (translated from Russian)

I. Grossman, W. Magnus, *Groups and Their Graphs, Random House* (The L. W. Singer Company, USA, 1964)

M.I. Kargapolov, Y.I. Merzlyakov, *Fundamentals of the Theory of Groups* (Springer, Germany, 1979). (translated from Russian)

M. Klin, G. Rücker, G. Tinhofer, *Algebraic Combinatorics in Mathematical Chemistry I. Methods and Algorithms. 1. Permutation Groups and Coherent (Cellular) Algebras* (Mathematical Institute, The Technical University of München, München, 1995)

J.S. Milne, *Group Theory*, Version 3.13, Copyright (2013)

G.R. Goodson, Inverse conjugacies and reversing symmetry groups. Am. Math. Mon. **106**(1), 19–26 (1999)

M. Baake, J.A.G. Roberts, Symmetries and reversing symmetries of polynomial automorphisms of the plane. ArXiv:math/0501151v1 [math.DS] (2005)

J. East, T.E. Nordahl, On groups generated by involutions of a semigroup. J. Algebra **445**, 136–162 (2016)

G. Targoński, On orbit theory and some of its applications. Zeszyty Nauk. Akad. Górn.-Hutniczej im. St. Staszica Nr. 764, Mat.-Fiz.-Chem **43**, 7–14 (1980)

V.R. Rosenfeld, Yet another generalization of Pólya’s theorem: enumerating equivalence classes of objects with a prescribed monoid of endomorphisms. MATCH Commun. Math. Comput. Chem. **43**, 111–130 (2001)

G.-C. Rota, D.A. Smith, Enumeration under group action. Ann. Sci. Norm. Super. Pisa. Cl. Sci. **4**, 637–646 (1977)

A. Pultr, Z. Herdlin, Relations (graphs) with given infinite semigroup. Monatsch. Math. **68**, 421–425 (1964)

Z. Herdlin, A. Pultr, Symmetric relations (undirected graphs) with given semigroup. Monatsch. Math. **69**, 318–322 (1965)

J. Sichler, Nonconstant endomorphisms of lattices. Proc. Am. Math. Soc. **34**(1), 67–70 (1972)

A. Pultr, V. Trnková, *Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories* (North-Holland Publishing Company, Amsterdam, 1980)

S. Mac Lane, *Categories for the Working Mathematician, Graduate Texts in Mathematics 5*, 2nd edn. (Springer, 1998). ISBN 0-387-98403-8. Zbl 0906.18001

G.B. Preston, Semigroups of graphs, in *Semigroups*, ed. by T.E. Hall, P.R. Jones, G.B. Preston (Academic Press, USA, 1980)

V.A. Molchanov, Semigroups on mappings of graphs. Semigroup Forum **27**(1–4), 155–199 (1983)

S.C. Shee, H.H. Teh, Graphical colour-representation of an inverse semigroup, in *Graph Theory, Proceedings of the 1st Southeast Asian Colloquium, Singapore 1983*, *Lecture Notes in Mathematics*, vol. 1073, pp. 222–227 (1984)

V. Koubek, V. Rödl, On the minimum order of graphs with given semigroup. J. Comb. Theory Ser. B **36**, 135–155 (1984)

L. Marki, Problems raised at the problem session on the colloquium on semigroups in Szeged, August 1987. Semigroup Forum, **37**(3), 367–373 (1988)

M.E. Adams, M. Gould, Posets whose monoids of order-preserving maps are regular. Order **6**, 195–201 (1989)

U. Knauer, M. Nieporte, Endomorphisms of graphs I. The monoid of strong endomorphisms. Archiv der Mathematik **52**(6), 607–614 (1989)

M. Böttcher, U. Knauer, Endomorphism spectra of graphs. Discrete Math. **109**, 45–57 (1992)

T.G. Lavers, The monoid of ordered partitions of a natural number. Semigroup Forum **53**, 44–56 (1996)

A. Solomon, Catalan monoids, monoids of local endomorphisms and their presentations. Semigroup Forum **53**, 351–368 (1996)

M.E. Adams, S. Bulman-Fleming, M. Gould, Endomorphism properties of algebraic structures, in: Proceedings of the Tennessee Topology Conference (World Scientific, 1997), pp. 1–17

W.M. Li, Inverses of regular strong endomorphisms of graphs. J. Math. Res. Exp. **18**(4), 529–534 (1998)

V.R. Rosenfeld, Endomorphisms of a weighted molecular graph and its spectrum. MATCH Commun. Math. Comput. Chem. **40**, 203–214 (1999)

T. Lavers, A. Solomon, The endomorphisms of a finite chain form a Rees congruence semigroup. Semigroup Forum **59**, 167–170 (1999)

M.E. Adams, M. Gould, Finite posets whose monoids of order-preserving maps are abundant. Acta Sci. Math. (Szeged) **67**, 3–37 (2001)

W. Li, J. Chen, Endomorphism-regularity of split graphs. Eur. J. Comb. **22**, 207–216 (2001)

G. Barnes, I. Levi, Ranks of semigroups generated by order-preserving transformations with a fixed partition type. Commun. Algebra **31**(4), 1753–1763 (2003)

W. Li, Graphs with regular monoids. Discrete Math. **265**, 105–118 (2003)

E. Sikolya, *Semigroups for flows in networks*, Doctoral thesis, the Eberhard-Karls University at Tübingen, 2004

P. Hell, *Graphs and Homomorphisms* (Oxford University Press, Oxford, 2004)

W.F. Klostermeyer, G. MacGillivray, Homomorphisms and oriented colorings of equivalence classes of oriented graphs. Discrete Math. **274**(1–3), 161–172 (2004)

W. Li, Various inverses of a strong endomorphism of a graph. Discrete Math. **300**, 245–255 (2005)

P. Ille, A proof of a conjecture of Sabidussi on graphs idempotent under the lexicographic product. Electron. Notes Discrete Math. **22**, 91–92 (2005)

S. Fan, Generalized symmetry of graphs. Electron. Notes Discrete Math. **23**, 51–60 (2005)

P.J. Cameron, Graph homomorphisms, in *Combinatorics Study Group Notes* (2006), pp. 1–7. www.maths.gmw.ac.uk/

P. Puusemp, Endomorphisms and endomorphism semigroups of groups, in *Focus on Group Theory Research*, ed. by L.M. Ying (Nova Science Publishers Inc, New York, 2006), pp. 27–57

A. Laradji, A. Umar, Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum **72**, 51–62 (2006)

A. Laradji, A. Umar, Combinatorial results for semigroups of order-preserving partial transformations. J. Algebra **278**, 342–359 (2006)

W. Mora, Y. Kemprasit, Regularity of full order-preserving transformation semigroups on some dictionary posets. Thai J. Math. **4**, 19–23 (2006). (Annual meeting in mathematics, 2006)

P.M. Higgins, J.D. Mitchell, M. Morayne, N. Ruškuc, Rank properties of endomorphisms of infinite partially ordered sets. Bull. Lond. Math. Soc. **38**, 177–191 (2006). doi:10.1112/S0024609305018138

E. Goode, D. Pixton, Recognizing splicing languages: syntactic monoids and simultaneous pumping. Discrete Appl. Math. **155**(8), 989–1006 (2007)

L. Kari, K. Mahalingam, G. Thierrin, The syntactic monoid of hairpin-free languages. Acta Inf. **44**, 153–166 (2007)

H. Hou, Y. Luo, Graphs whose endomorphism monoids are regular. Discrete Math. **308**, 3888–3896 (2008)

H. Hou, Y. Luo, Z. Cheng, The endomorphism monoid of (overline<>**29**, 1173–1185 (2008)

Sr Arworn, An algorithm for the numbers of endomorphisms on paths (DM13208). Discrete Math. **309**, 94–103 (2009)

V. Koubek, V. Rödl, B. Shemmer, On graphs with a given endomorphism monoid. J. Graph Theory **62**, 241–262 (2009)

J.D. Mitchell, M. Morayne, Y. Péresse, M. Quick, Generating transformation semigroups using endomorphisms of preoders, graphs, and tolerances. Ann. Pure Appl. Logic **161**(12), 1471–1485 (2010)

R. Kaschek, On wreathed lexicographic products of graphs. Discrete Math. **310**, 1275–1281 (2010)

C. Manon, Presentations of semigroup algebras of weighted trees. J. Algebraic Comb. **31**, 467–489 (2010)

Y. Kemprasit, W. Mora, T. Rungratgasame, Isomorphism theorems for semigroups of order-preserving partial transformations. Int. J. Algebra **4**(17), 799–808 (2010)

V.H. Fernandes, M.M. Jesus, V. Maltcev, J.D. Mitchell, Endomorphisms of the semigroup of order-preserving mappings. Semigroup Forum **81**, 277–285 (2010)

J. Araújo, M. Kinyonc, J. Konieczny, Minimal paths in the commuting graphs of semigroups. Eur. J. Comb. **32**, 178–197 (2011)

E.A. Bondar’, V. Yu Zhuchok, Representations of the monoid of strong endomorphisms of (n) -uniform hypergraphs. Fund. Appl. Math. **18**(1), 21–34 (2013). (in Russian)

D. Roberson, *Variations on a Theme: Graph Homomorphisms*, Doctoral thesis, the University of Waterloo, 2013

W. Buczynska, J. Buczynski, K. Kubjas, M. Michalek, On the graph labellings arising from phylogenetics. Cent. Eur. J. Math. **11**(9), 1577–1592 (2013)

H. Hinton, *Theory of Groups of Finite Order* (Oxford University Press, Oxford, 1908)

A.H. Clifford, G.B. Preston, *The Algebraic Theory of Semigroups*, vol. 1 (American Mathematical Society, Providence, 1961)

A.H. Clifford, G.B. Preston, *The Algebraic Theory of Semigroups*, vol. 2 (American Mathematical Society, Providence, 1967)

P.M. Higgins, *Techniques of Semigroup Theory* (Oxford University Press, Oxford, 1992)

J.M. Howie, *An Introduction to Semigroup Theory* (Academic Press, USA, 1976)

J.M. Howie, *Fundamentals of Semigroup Theory* (Oxford University Press, Oxford, 1995)

G. Lallement, *Semigroups and Combinatorial Applications* (Wiley, USA, 1979)

E.S. Lyapin, *Semigroups* (American Mathematical Society, Providence, 1974) Translations of Mathematical Monographs, vol. 3

M. Petrich, *Introduction to Semigroups* (Merrill, USA, 1973)

M. Petrich, *Lectures in Semigroups* (Wiley, USA, 1977)

M. Petrich, *Inverse Semigroups* (Wiley, USA, 1984)

M. Petrich, N.R. Reilly, *Completely Regular Semigroups (Canadian Mathematicsal Society Series of Monographs and Advanced Texts)* (Wiley, New York, 1999)

F.J. Pastijn, M. Petrich, *Regular Semigroups as Extensions*, Ser.: Research Notes in Mathematics, Vol. 136, Pitman Publishing Program, Boston, 1985

R.H. Bruck, *A Survey of Binary Systems* (Springer, New York, 1971)

L.N. Shevrin, Semigroups, in *General Algebra*, vol. 2, ed. by L.A. Skornyakov (Nauka, Moscow, 1991), pp. 11–191

J.E. Pin, Finite semigroups and recognizable languages, in *Semigroups, Formal Languages and Groups*, ed. by J. Fountain (Kluwer Academic Publishers, Dodrecht, 1995)

L.N. Shevrin, A.J. Ovsyannikov, *Semigroups and Their Subsemigroup Lattices* (Kluwer Academic Publishers, USA, 1996)

A.L.T. Paterson, *Groupoids, Inverse Semigroups, and Their Operator Algebras*, Ser.: Progress in Mathematics, Vol. 170, Birkhäuser, Boston, USA, 1998

T. Harju, *Lecture Notes on Semigroups* (University of Turku, Turku, 1996)

S.A. Duplij, *Polusupermnogoobraziya i polugruppy = Semisupermanifolds and Semigroups* (Krok, Kharkov, 2000 and 2013) **(in Russian)**

T.S. Blyth, M.H. Almeida Santos, Regular semigroups with skew pairs of idempotents. Semigroup Forum **65**, 264–274 (2002)

V.R. Rosenfeld, Combinatorial actions of idealizers of subsemigroups (submitted)

A.C. Brown, On an application of mathematics to chemistry. Proc. R. Soc. Edinb. **6**(73), 89–90 (1866–1867)

V. R. Rosenfeld and Victor R. Rosenfeld, “Groupoids and classification of polymerization reactions”, in: *The Use of Computers in Spectroscopy and Chemical Research, Novosibirsk, 6–8th September 1983, Theses of the All-Union Conference*, Novosibirsk, USSR, 1983, pp. 195–196 **(in Russian)**

V.R. Rosenfeld, Using semigroups in modeling of genomic sequences. MATCH Commun. Math. Comput. Chem. **56**(2), 281–290 (2006)

J.-A. de Séguier, *Élements de la Théorie des Groupes Abstraits (Elements of the Theory of Abstract Groups)* (Gauthier-Villars, Paris, 1904)

K. Fichtner, On groupoids in crystallography. MATCH Commun. Math. Comput. Chem. **9**, 21–40 (1980)

A. Weinstein, Groupoids: unifying internal and external symmetry. Not. Am. Math. Soc. **43**(7), 744–752 (1996)

R.T. Živaljević, Combinatorial groupoids, cubical complexes, and the Lovász conjecture. arXiv:math/0510204v2 [math.CO] (2005)

R.T. Živaljević, Groupoids in combinatorics—applications of a theory of local symmetries. arXiv:math/0605508v1 [math.CO] (2006)

I. Sciriha, S. Fiorini, On the characteristic polynomial of homeomorphic images of a graph. Discrete Math. **174**, 293–308 (1997)

M.V. Diudea, I. Gutman, L. Jantschi, *Molecular Topology* (Nova Science Publishers, New York, 2001)

V.R. Rosenfeld, Equivalent genomic (proteomic) sequences and semigroups. J. Math. Chem. **53**(6), 1488–1494 (2015). doi:10.1007/s10910-015-0501-y

V.D. Belousov, *Foundations of the Theory of Quasigroups and Loops* (USSR, Moscow, 1967). (in Russian)

V.D. Belousov, *Elements of Quasigroup Theory: A Special Course* (Kishinev State University Printing House, Kishinev, 1981). (in Russian)

H.O. Pflugfelder, *Quasigroups and Loops: Introduction* (Heldermann, Berlin, 1990)

H.O. Pflugfelder, Historical notes on loop theory. Comment. Math. Univ. Carol. **41**(2), 359–370 (2000)

## AES E-Library

, "Complete Journal: Volume 46 Issue 7/8," *J. Audio Eng. Soc.*, vol. 46, no. 7/8, (1998 July/August.). doi: , "Complete Journal: Volume 46 Issue 7/8," *J. Audio Eng. Soc.*, vol. 46 Issue 7/8, (1998 July/August.). doi:

Abstract: This is a complete Journal issue.

TY - paper

TI - Complete Journal: Volume 46 Issue 7/8

SP - EP -

PY - 1998

JO - Journal of the Audio Engineering Society

IS - 7/8

VO - 46

VL - 46

Y1 - July/August 1998 TY - paper

TI - Complete Journal: Volume 46 Issue 7/8

SP - EP -

PY - 1998

JO - Journal of the Audio Engineering Society

IS - 7/8

VO - 46

VL - 46

Y1 - July/August 1998

AB - This is a complete Journal issue.

This is a complete Journal issue.

JAES Volume 46 Issue 7/8 July/August 1998

Publication Date: July 1, 1998 Import into BibTeX

Permalink: http://www.aes.org/e-lib/browse.cfm?elib=19136

Click to purchase paper as a non-member or login as an AES member. If your company or school subscribes to the E-Library then switch to the institutional version. If you are not an AES member and would like to subscribe to the E-Library then Join the AES!

*This paper costs $33 for non-members and is free for AES members and E-Library subscribers.*