Hecke algebra characters and immanant conjectures

Haiman, Mark

doi:10.1090/S0894-0347-1993-1186961-9

Hecke algebra characters and immanant conjectures

Author: Mark Haiman
Journal: J. Amer. Math. Soc. 6 (1993), 569-595
MSC: Primary 05E05; Secondary 14M15, 20C30, 20F55
DOI: https://doi.org/10.1090/S0894-0347-1993-1186961-9
MathSciNet review: 1186961
Full-text PDF Free Access

Dan Barbasch and David Vogan, Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra 80 (1983), no. 2, 350–382. MR 691809, DOI https://doi.org/10.1016/0021-8693%2883%2990006-6
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
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Francesco Brenti, Unimodal polynomials arising from symmetric functions, Proc. Amer. Math. Soc. 108 (1990), no. 4, 1133–1141. MR 993741, DOI https://doi.org/10.1090/S0002-9939-1990-0993741-2
J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387–410. MR 632980, DOI https://doi.org/10.1007/BF01389272
P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 239–246. MR 776477, DOI https://doi.org/10.1090/S0273-0979-1985-15361-3
Mark Goresky and Robert MacPherson, Intersection homology. II, Invent. Math. 72 (1983), no. 1, 77–129. MR 696691, DOI https://doi.org/10.1007/BF01389130
I. P. Goulden and D. M. Jackson, Immanants of combinatorial matrices, J. Algebra 148 (1992), no. 2, 305–324. MR 1163738, DOI https://doi.org/10.1016/0021-8693%2892%2990196-S
Curtis Greene, Proof of a conjecture on immanants of the Jacobi-Trudi matrix, Linear Algebra Appl. 171 (1992), 65–79. MR 1165445, DOI https://doi.org/10.1016/0024-3795%2892%2990250-E
Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. MR 0143793
Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943
Michio Jimbo, A $q$-analogue of $U({\mathfrak g}{\mathfrak l}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247–252. MR 841713, DOI https://doi.org/10.1007/BF00400222
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI https://doi.org/10.2307/1971403
V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, DOI https://doi.org/10.1007/BF01389127
A. Joseph, Towards the Jantzen conjecture. I, II, Compositio Math. 40 (1980), no. 1, 35–67, 69–78. MR 558258
David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI https://doi.org/10.1007/BF01390031
David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185–203. MR 573434
R. C. King and B. G. Wybourne, Characters of Hecke algebras $H_n(q)$ of type $A_{n-1}$, J. Phys. A 23 (1990), no. 23, L1193–L1197. MR 1086259
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
Elliott H. Lieb, Proofs of some conjectures on permanents, J. Math. Mech. 16 (1966), 127–134. MR 0202745, DOI https://doi.org/10.1512/iumj.1967.16.16008

D. E. Littlewood, The theory of group characters, 2nd ed., Oxford Univ. Press, London, 1950.

I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. MR 553598
Arun Ram, A Frobenius formula for the characters of the Hecke algebras, Invent. Math. 106 (1991), no. 3, 461–488. MR 1134480, DOI https://doi.org/10.1007/BF01243921
I. Schur, Über endliche Gruppen und Hermitesche Formen, Math. Z. 1 (1918), no. 2-3, 184–207 (German). MR 1544291, DOI https://doi.org/10.1007/BF01203611
M.-P. Schützenberger, La correspondance de Robinson, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Springer, Berlin, 1977, pp. 59–113. Lecture Notes in Math., Vol. 579 (French). MR 0498826

T. A. Springer, Quelques applications de la cohomologie d’intersection, Sém. Bourbaki 589 (1982).

Richard P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986) Ann. New York Acad. Sci., vol. 576, New York Acad. Sci., New York, 1989, pp. 500–535. MR 1110850, DOI https://doi.org/10.1111/j.1749-6632.1989.tb16434.x
Richard P. Stanley and John R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A 62 (1993), no. 2, 261–279. MR 1207737, DOI https://doi.org/10.1016/0097-3165%2893%2990048-D
J. R. Stembridge, Some conjectures for immanants, Canad. J. Math. 44 (1992), no. 5, 1079–1099. MR 1186482, DOI https://doi.org/10.4153/CJM-1992-066-1
John R. Stembridge, Eulerian numbers, tableaux, and the Betti numbers of a toric variety, Discrete Math. 99 (1992), no. 1-3, 307–320. MR 1158793, DOI https://doi.org/10.1016/0012-365X%2892%2990378-S
John R. Stembridge, Immanants of totally positive matrices are nonnegative, Bull. London Math. Soc. 23 (1991), no. 5, 422–428. MR 1141010, DOI https://doi.org/10.1112/blms/23.5.422
J. Van der Jeugt, An algorithm for characters of Hecke algebras $H_n(q)$ of type $A_{n-1}$, J. Phys. A 24 (1991), no. 15, 3719–3725. MR 1123644
Hans Wenzl, Hecke algebras of type $A_n$ and subfactors, Invent. Math. 92 (1988), no. 2, 349–383. MR 936086, DOI https://doi.org/10.1007/BF01404457