Clifford-symmetric polynomials

TL;DR

This paper introduces the Clifford Hecke superalgebra as a new super-algebraic analogue of the NilHecke algebra, exploring symmetric polynomials within this framework and their fundamental properties.

Contribution

It develops the Clifford Hecke superalgebra and establishes a theory of symmetric polynomials analogous to elementary symmetric polynomials in this new setting.

Findings

Definition of Clifford Hecke superalgebra $\

Identification of symmetric polynomials within this algebraic framework

Proof that these symmetric polynomials are generated by an analogue of elementary symmetric polynomials

Abstract

Based on the NilHecke algebra $\mathsf{NH}_n$, the odd NilHecke algebra developed by Ellis, Khovanov and Lauda and Kang, Kashiwara and Tsuchioka's quiver Hecke superalgebra, we develop the Clifford Hecke superalgebra $\mathsf{NH}\mathfrak{C}_n$ as another super-algebraic analogue of $\mathsf{NH}_n$. We show that there is a notion of symmetric polynomials fitting in this picture, and we prove that these are generated by an appropriate analogue of elementary symmetric polynomials, whose properties we shall discuss in this text.