The q-Analogue of the Alternating Group and Its Representations

Abstract

We define the new algebra. This algebra has a parameter q. The defining relations of this algebra at q s 1 coincide with the basic relations of the alternating group. We also give the new subalgebra of the Hecke algebra of type A which is isomorphic to this algebra. This algebra is free of rank half that of the Hecke algebra. Hence this algebra is regarded as a q-analogue of the alternating group.All the isomorphism classes of the irreducible representations of this algebra and the q-analogue of the branching rule between the symmetric group and the alternating group are obtained. ᮊ

Quiver Hecke algebras for alternating groups

“…These algebras include, as special cases, the group algebras of the symmetric group and the Iwahori-Hecke algebras of type A. This paper extends these results to the group algebras of the alternating groups and, more generally, to Mitsuhashi's alternating Hecke algebras [20].…”

“…Since the group algebra of the alternating group is the fixed-point subalgebra of the sign automorphism, the following definition gives a ξ-analogue of the group ring ZA n of the alternating group A n . [20]). Suppose that ξ = −1.…”

The local stationary presentation of the alternating groups and the normal form

“…First of all it opens the way of construction of the representation theory of alternating groups in the spirit of Okounkov-Vershik approach [6,10] independently of the representation theory of the symmetric groups; this means-to define the analogues of Gelfand-Zetlin algebra, Young-Jucys-Murphy elements, etc. Secondly, our presentation of the alternating groups allows to give various generalizations of alternating groups like anticommuting (Z 2 or fermionic version (see [11])), q-analogs (see also [5] 3 ).…”

Schur–Weyl Reciprocity between the Quantum Superalgebra and the Iwahori–Hecke Algebra

“…In [9], we defined a q-deformation of the alternating group as a subalgebra of the Iwahori-Hecke algebra, and determined all the isomorphism classes of (ordinary) irreducible representations. After [9], we intended to compute character values of irreducible representations directly using combinatorial methods.…”

“…After [9], we intended to compute character values of irreducible representations directly using combinatorial methods. But this approach did not go well because the notion of conjugacy classes of the q-deformation of the alternating group is obscure, hence we could not apply the classical(q = 1) case which is found in [5].…”