Sandwich matrices, Solomon algebras, and Kazhdan-Lusztig polynomials

Putcha, Mohan S.

doi:10.1090/S0002-9947-1993-1127157-2

Sandwich matrices, Solomon algebras, and Kazhdan-Lusztig polynomials
HTML articles powered by AMS MathViewer

by Mohan S. Putcha PDF

Trans. Amer. Math. Soc. 340 (1993), 415-428 Request permission

Abstract:

Sandwich matrices have proved to be of importance in semigroup theory for the last 50 years. The work of the author on algebraic monoids leads to sandwich matrices in group theory. In this paper, we find some connections between sandwich matrices and the Hecke algebras (for monoids) introduced recently by Louis Solomon. At the local level we then obtain an explicit isomorphism between Solomon’s Hecke algebra and the complex monoid algebra of the Renner monoid. In the simplest case of monoids associated with a Borel subgroup, we find that the entries of the inverse of the sandwich matrix, as well as those of the related structure matrix of Solomon’s Hecke algebra are ’almost’ the polynomials ${R_{x,y}}$ associated with the Kazhdan-Lusztig polynomials.

References

Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
A. H. Clifford, Matrix representations of completely simple semigroups, Amer. J. Math. 64 (1942), 327–342. MR 6551, DOI 10.2307/2371687
A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499–511. MR 782232, DOI 10.1007/BF01388520
Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), no. 2, 483–506. MR 916182, DOI 10.1016/0021-8693(87)90232-8
J. A. Green, On the structure of semigroups, Ann. of Math. (2) 54 (1951), 163–172. MR 42380, DOI 10.2307/1969317
Nagayoshi Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 215–236 (1964). MR 165016
David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
W. D. Munn, On semigroup algebras, Proc. Cambridge Philos. Soc. 51 (1955), 1–15. MR 66355, DOI 10.1017/s0305004100029868
W. D. Munn, The characters of the symmetric inverse semigroup, Proc. Cambridge Philos. Soc. 53 (1957), 13–18. MR 81910, DOI 10.1017/s0305004100031947
Jan Okniński and Mohan S. Putcha, Complex representations of matrix semigroups, Trans. Amer. Math. Soc. 323 (1991), no. 2, 563–581. MR 1020044, DOI 10.1090/S0002-9947-1991-1020044-8
Jan Okniński and Mohan S. Putcha, Parabolic subgroups and cuspidal representations of finite monoids, Internat. J. Algebra Comput. 1 (1991), no. 1, 33–47. MR 1112298, DOI 10.1142/S0218196791000031
I. S. Ponizovskiĭ, On matrix representations of associative systems, Mat. Sb. N.S. 38(80) (1956), 241–260 (Russian). MR 0081292
Mohan S. Putcha, A semigroup approach to linear algebraic groups, J. Algebra 80 (1983), no. 1, 164–185. MR 690712, DOI 10.1016/0021-8693(83)90026-1
Mohan S. Putcha, Determinant functions on algebraic monoids, Comm. Algebra 11 (1983), no. 7, 695–710. MR 694597, DOI 10.1080/00927878308822873
Mohan S. Putcha, Monoids on groups with $BN$-pairs, J. Algebra 120 (1989), no. 1, 139–169. MR 977865, DOI 10.1016/0021-8693(89)90193-2
Mohan S. Putcha, Monoids of Lie type and group representations, Monoids and semigroups with applications (Berkeley, CA, 1989) World Sci. Publ., River Edge, NJ, 1991, pp. 288–305. MR 1142383
Mohan S. Putcha, Classification of monoids of Lie type, J. Algebra 163 (1994), no. 3, 636–662. MR 1265855, DOI 10.1006/jabr.1994.1035
M. S. Putcha and L. E. Renner, The system of idempotents and the lattice of ${\scr J}$-classes of reductive algebraic monoids, J. Algebra 116 (1988), no. 2, 385–399. MR 953159, DOI 10.1016/0021-8693(88)90225-6
Mohan S. Putcha and Lex E. Renner, The canonical compactification of a finite group of Lie type, Trans. Amer. Math. Soc. 337 (1993), no. 1, 305–319. MR 1091231, DOI 10.1090/S0002-9947-1993-1091231-X
D. Rees, On semi-groups, Proc. Cambridge Philos. Soc. 36 (1940), 387–400. MR 2893
Lex E. Renner, Analogue of the Bruhat decomposition for algebraic monoids, J. Algebra 101 (1986), no. 2, 303–338. MR 847163, DOI 10.1016/0021-8693(86)90197-3
Lex E. Renner, Finite monoids of Lie type, Monoids and semigroups with applications (Berkeley, CA, 1989) World Sci. Publ., River Edge, NJ, 1991, pp. 278–287. MR 1142382
Louis Solomon, The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field, Geom. Dedicata 36 (1990), no. 1, 15–49. MR 1065211, DOI 10.1007/BF00181463

Reductive monoids