On representation theory of cyclotomic Hecke-Clifford algebras

TL;DR

This paper explicitly constructs simple modules for cyclotomic Hecke-Clifford superalgebras and provides conditions for their semi-simplicity, demonstrating that these algebras are generically semisimple.

Contribution

It offers a new explicit construction of simple modules and establishes criteria for semi-simplicity in cyclotomic Hecke-Clifford superalgebras.

Findings

Explicit simple module construction for non-degenerate and degenerate cases

Sufficient conditions for semi-simplicity based on dimension comparison

Proof that generic cyclotomic Hecke-Clifford superalgebras are semisimple

Abstract

In this article, we give an explicit construction of the simple modules for both non-degenerate and degenerate cyclotomic Hecke-Clifford superalgebras over an algebraically closed field of characteristic not equal to $2$ under certain condition in terms of parameters in defining these algebras. As an application, we obtain a sufficient condition on the semi-simplicity of these cyclotomic Hecke-Clifford superalgebras via a dimension comparison. As a byproduct, both generic non-degenerate and degenerate cyclotomic Hecke-Clifford superalgebras are shown to be semisimple.