Abstract
By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. En étudiant l'action de l'algèbre de 0-Hecke sur l'algèbre coinvariante et la variété de drapeaux complète, nous interprétons les fonctions génératrices qui comptent les permutations avec un ensemble inverse de descentes fixé, selon leur nombre d'inversions et leur "major index''.
Highlights
A composition I of an integer n gives rise to a descent class of permutations in the symmetric group Sn; the cardinality of this descent class is known as the ribbon number rI and its inv-generating function is the q-ribbon number rI (q)
The 0-Hecke algebra Hn(0) acts on the coinvariant algebra Z[x]/(Z[x]S+n ) via the operators πi = πi − 1, where πi is the Demazure operator defined by πif xif − si(xif ) . xi − xi+1
It is well known that compositions of n bijectively correspond to the subsets of [n − 1] via their descent set; they bijectively correspond to the ribbon diagrams, i.e. connected skew Young diagrams without 2 × 2 boxes, whose row sizes from bottom to top are i1, . . . , in
Summary
A composition I of an integer n gives rise to a descent class of permutations in the symmetric group Sn; the cardinality of this descent class is known as the ribbon number rI and its inv-generating function is the q-ribbon number rI (q). Reiner and Stanton [15] defined a (q, t)-ribbon number rI (q, t), and gave an interpretation by representations of Sn and GL(n, Fq). Our main object here is to obtain similar interpretations of various ribbon numbers by representations of the 0-Hecke algebra Hn(0) of type A. Norton [14] decomposed Hn(0) into a direct sum of 2n−1 distinct indecomposable Hn(0)-submodules MI indexed by compositions I of n. Every indecomposable projective Hn(0)-module is isomorphic to MI for some I, and every simple Hn(0)module is isomorphic to CI = top(MI ) = MI /rad MI for some I
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