A super Frobenius formula for the characters of Iwahori–Hecke algebras

Abstract

In this paper, we establish a super Frobenius formula for the characters of Iwahori-Hecke algebras. We define Hall-Littlewood supersymmetric functions in a standard manner to make supersymmetric functions from symmetric functions, and investigate some properties of supersymmetric functions. Based on Schur-Weyl reciprocity between Iwahori-Hecke algebras and the general quantum super algebras, which was obtained in [8], we derive that one of several types of Hall-Littlewood supersymmetric functions, up to constant, generates the values of the irreducible characters of Iwahori-Hecke algebras at the elements corresponding to cycle permutations. Our formula in this article includes both the ordinary quantum case (n = 0) that was obtained in [10] and the classical super case (q→1).

Schur-Sergeev duality for Ariki-Koike algebras

“…between the universal enveloping superalgebra U(g) of g and the group algebra CW m,n of W m,n , which is a generalization of the Schur-Sergeev duality in [5,40]. (ii) Based on Shoji's work [42] and Mitsuhashi's work [29], we will give a super Frobenius formula for H in [46], which is one of our motivation to construct the Schur-Weyl duality between quantum superalgebras and cyclotomic Hecke algebras. (iii) In the forthcoming work, we will introduce the cyclotomic q-Schur superalgebras, which enable us to give an alternative proof of Theorem 5.13.…”

“…Based on the quantum Schur-Weyl reciprocity, Ram [32] gave a q-analogue of Frobenius formula for the characters of the Iwahori-Hecke algebras of type A. A super Frobenius formula for the characters of the Iwahori-Hecke algebras of type A was given by Mitsuhashi in [29] by applying the super quantum Schur-Weyl reciprocity. An extension of Frobenius formula for the characters of cyclotomic Hecke algebra of type G(m, 1, n) is found in [42] by applying the Schur-Weyl reciprocity between cyclotomic Hecke algebras and quantum algebras given in [34].…”

“…Motivated by these works, a natural problem is to provide a cyclotomic (quantum) analogue of these formulas, which is another motivation of the present paper. Base on [29,42], we will give a super Frobenius formula in [46] and a cyclotomic (quantum) analogue of Regev formula for the characters of cyclotomic Hecke algebras in a forthcoming article.…”

A super Frobenius formula for the characters of cyclotomic Hecke algebras

“…Following [10], we define the super Hall-Liitlewood function q a (x/y; t) ∈ (Λ k,ℓ ) Q(t) as follows:…”

“…Based on the Schur-Weyl reciprocity between the Iwahori-Hecke algebras of type A and the quantum enveloping algebra of gl(n) given by Jimbo [7], Ram [12] gave a q-analogue of Frobenius formula for the characters of the Iwahori-Hecke algebras of type A. A super Frobenius formula for the characters of the Iwahori-Hecke algebras of type A was given by Mitsuhashi in [10] by virtue of the Schur-Weyl reciprocity between the Iwahori-Hecke algebras of type A and the quantum superalgebra [9,11]. An extension of Frobenius formula for the characters of cyclotomic Hecke algebra of type G(m, 1, n) is found in [17] by applying the Schur-Weyl reciprocity between cyclotomic Hecke algebras and quantum algebras given in [13].…”