R-Polynomials of Finite Monoids of Lie Type

Abstract

This paper studies the combinatorics of the orbit Hecke algebras associated with W × W orbits in the Renner monoid of a finite monoid of Lie type, M, where W is the Weyl group associated with M. It is shown by Putcha in [12] that the Kazhdan–Lusztig involution [6] can be extended to the orbit Hecke algebra which enables one to define the R-polynomials of the intervals contained in a given orbit. Using the R-polynomials, we calculate the Möbius function of the Bruhat–Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group.

The rook monoid is lexicographically shellable

“…is k, by our induction hypothesis, it is the lexicographically first chain between x ′ 1 and y. We will find contradictions in each of the following (exhaustive) possibilities to conclude that the c ′ does not increase, therefore, c is unique: Let R n,k ⊂ R n (0 ≤ k ≤ n) denote the subposet consisting of elements whose rank is k. In [1], it is shown that the Möbius function on I(R n,k ) takes values in {−1, 0, 1}. When k = n, R n,k is the symmetric group, and the Möbius function on S n is well known (see [19,18,9]).…”

“…We denote by I(P ) the set of all intervals in P . The Möbius function µ : I(P ) −→ Z is an integer valued function, (uniquely) determined by the following conditions: Let R n,k ⊂ R n (0 ≤ k ≤ n) denote the subposet consisting of elements whose rank is k. In [1], it is shown that the Möbius function on I(R n,k ) takes values in {−1, 0, 1}. When k = n, R n,k is the symmetric group, and the Möbius function on S n is well known (see [19,18,9]).…”

“…We denote by C(P ) the set of pairs (x, y) from P × P such that y covers x. The poset P is called lexicographically shellable, or EL-shellable, if there exists a map f : C(P ) → [n] such that (1) in every interval [x, y] ⊆ P , there exists a unique maximal chain c : x = x 0 < x 1 < · · · < x k+1 = y such that f (x i , x i+1 ) ≤ f (x i+1 , x i+2 ) for i = 0, . .…”