The H-polynomial of an irreducible representation

Abstract

Let G be a simple algebraic group. Associated with the finite-One can identify the cases where P ρ is rationally smooth; and in such cases it is desirable to calculate the H-polynomial, H, of P ρ . In this paper we consider the situation where ρ is irreducible. We then determine H explicitly in terms of combinatorial invariants of ρ. Indeed, there is a canonical cellular decomposition for P ρ . These cells are defined in terms of idempotents, B × B-orbits and other natural quantities obtained from M ρ .Furthermore, H is obtained by recording the dimension of each of these cells in terms of the descent system of M ρ . As a special case we reacquire the well-known formula for the Poincaré polynomial of a "wonderful embedding" of a simple algebraic group of adjoint type.

The H-polynomial of a Group Embedding

“…See [12] for a detailed discussion of J-irreducible monoids. See [23] for a detailed account of the results of this section. The result that gets us going here is the following.…”

“…In this section we discuss combinatorially smooth subsets J of S. These subsets correspond to the rationally smooth G × G-embeddings X ρ of a semisimple group G with ρ an irreducible representation. The formulas of this subsection are the culmination of the results of [19,20,21,22,23].…”

“…Throughout this survey we omit the proofs of many statements. The interested reader should consult [19,20,21,22,23] for more details.…”

A Geometric Interpretation of the Intertwining Number

“…One can think of the H-polynomial of R as a transformed "Hasse-Weil motivic zeta function" of the projectivization P(M − {0}) of M. Indeed, by treating q as a power of a prime number, for all sufficiently large q, (H(R, q) − 1)/(q − 1) is equal to the number of rational points over F q (the finite field with q elements) of P(M − {0}). See [18,Remark 3.2]. For an application of this idea to the rook theory, we recommend [7].…”

Equivariant Cohomology of Rationally Smooth Group Embeddings

“…But even more is true. By [R7,Corollary 2.3] the face lattice F λ of P λ is completely described in terms of the Weyl group (W, Σ). Indeed, the set of W -orbits of F λ is in one-to-one correspondance with {I ⊆ Σ | no connected component of I is contained entirely in J}.…”