Representations of Hecke-Clifford superalgebras at roots of unity

TL;DR

This paper classifies irreducible representations of affine Hecke-Clifford superalgebras at roots of unity and determines when finite Hecke-Clifford superalgebras are semisimple based on the root order.

Contribution

It provides a complete classification of irreducible splittable representations at roots of unity and characterizes semisimplicity conditions for finite Hecke-Clifford superalgebras.

Findings

Irreducible splittable representations are classified at roots of unity.

H_n(q) is semisimple if and only if h > n (odd h) or h > 2n (even h).

Semisimplicity depends on the order of the root of unity.

Abstract

In this article, we give a classification of irreducible completely splittable representations of affine Hecke-Clifford superalgebras $H_n^{\mathrm{aff}}(q)$ when $q^2$ is a primitive $h$-th root of unity. As an application, we derive a necessary and sufficient condition for the finite Hecke-Clifford superalgebra $H_n(q)$ to be semisimple. Specially we show that $H_n(q)$ is semisimple if and only $h >n$ in the case $h$ is odd and $h >2n$ in the case $h$ is even.