0-Hecke algebra actions on coinvariants and flags

Abstract

The 0-Hecke algebra Hn(0) is a deformation of the group algebra of the symmetric group Sn. We show that its coinvariant algebra naturally carries the regular representation of Hn(0), giving an analogue of the well-known result for Sn by Chevalley-Shephard-Todd. By investigating the action of Hn(0) on coinvariants and flag varieties, we interpret the generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. We also study the action of Hn(0) on the cohomology rings of the Springer fibers, and similarly interpret the (noncommutative) Hall-Littlewood symmetric functions indexed by hook shapes. P α summed over all compositions α of n, where every P α is a (left) indecomposable H n (0)-module. It follows that {P α : α |= n} is a complete list of non-isomorphic projective indecomposable H n (0)modules, and {C α : α |= n} is a complete list of non-isomorphic simple H n (0)-modules, where C α = top(P α ) = P α / rad P α for all compositions α |= n. for their helpful suggestions and comments.

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

“…It is intimately related with the Hopf algebras of quasisymmetric functions and noncommutative symmetric functions respectively [13], in the same way as symmetric group representation theory is intimately connected with the Hopf algebra of symmetric functions. More information about H n (0) and its representations can be found in [10,34], and contemporary results can be found in [6,23,24,25,26,29]. Our interest in the 0-Hecke algebra stems from the authors' previous work in the context of providing a representation-theoretic interpretation for quasisymmetric Schur functions [45].…”

Recent Trends in Quasisymmetric Functions

“…Norton [96] investigates the representation theory of H n (0) and proves that there are 2 n−1 distinct irreducible representations of H n (0), indexed by compositions of n. Let G 0 (H n (0)) be the Grothendieck group of finitely generated H n (0)-modules and G = ⊕ n≥0 G 0 (H n (0)) be the associated Grothendieck ring. (See Carter [23] for a thorough account of the representation theory of the 0-Hecke algebra and see Huang [71,72,73] for recent connections with flag varieties, the Stanley-Reisner ring, and tableaux.) Krob and Thibon [76] prove that G is isomorphic to the ring of quasisymmetric functions via a characteristic map F : G → QSym called the quasisymmetric characteristic.…”

Tail Positive Words and Generalized Coinvariant Algebras

“…The space Λ of symmetric functions has many generalizations; in this paper we will also use the spaces QSym of quasisymmetric functions and NSym of noncommutative symmetric functions. We briefly review their definition below, as well as their relationship with the 0-Hecke algebra H n (0); for more details see [12,13]. Let n be a positive integer.…”

“…The indecomposable projective representations of H n (0) are naturally labeled by compositions α |= n (see [12,13]). For α |= n, we let P α denote the corresponding indecomposable projective and let (2.9) 0)) is free with basis given by (isomorphism classes of) the irreducibles {C α : α |= n}.…”