Quiver Hecke algebras for alternating groups

TL;DR

This paper establishes that alternating Hecke algebras over sufficiently large fields of characteristic not 2 are $Z$-graded and isomorphic to fixed-point subalgebras of quiver Hecke algebras, providing new structural insights.

Contribution

It introduces a homogeneous presentation and proves that these algebras are graded symmetric, extending the understanding of their structure and relations to quiver Hecke algebras.

Findings

Alternating Hecke algebras are $Z$-graded over large enough fields.

They are isomorphic to fixed-point subalgebras of quiver Hecke algebras.

Blocks of these algebras are graded symmetric.

Abstract

The main result of this paper shows that, over large enough fields of characteristic different from $2$, the alternating Hecke algebras are $\mathbb{Z}$-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke algebra of the symmetric group $\mathfrak{S}_n$. As a special case, this shows that the group algebra of the alternating group, over large enough fields of characteristic different from $2$, is a $\mathbb{Z}$-graded algebra. We give a homogeneous presentation for these algebras, compute their graded dimension and show that the blocks of the quiver Hecke algebras of the alternating group are graded symmetric algebras.