The H-polynomial of a semisimple monoid

Abstract

A semisimple monoid M is called quasismooth if M \ {0} has sufficiently mild singularities. We define a cellular decomposition of such monoids using the method of one-parameter subgroups. These cells turn out to be "almost" affine spaces. But they can also be described in terms of the idempotents and B × B-orbits of M. This leads to a number of combinatorial results about the inverse monoid of B × B-orbits of M. In particular, we obtain fundamental information about the H -polynomial of M.

The H-polynomial of an irreducible representation

“…However, in [17], we were not yet able to completely quantify dim(C r ) and r → f r in terms of the descent system (W J , S J ). In Theorem 5.5 of [17] we did obtain much information about the dimension of each cell C r .…”

“…But these cellular decompositions can often be obtained in the presence of mild singularities. See [15,17]. One major purpose of this paper …”

“…An h-polynomial can be seen as the special case of an H -polynomial where B = T . In [17] we investigated the H -polynomial, H(M), of a reductive monoid M. We showed how to calculate H(M) in terms of more basic geometric quantities. In particular we reduced the calculation of dim(C r ) to the problem of understanding a certain idempotent f r with the property…”

“…In Theorem 5.5 of [17] we did obtain much information about the dimension of each cell C r . This led us to the following formula for H(M).…”

Computation in Linear Algebraic Monoids

“…Recently, in [1,[16][17][18] Can and Renner gave a systematic description of H-polynomials of reductive monoids, whose whole point is to investigate the orders of finite reductive monoids.…”