Clifford-symmetric polynomials

Abstract

Based on the NilHecke algebra $\mathsf{N}{\mathsf{H}}_n$, the odd NilHecke algebra developed by Ellis, Khovanov and Lauda, and on 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 $\mathsf{N}{\mathsf{H}}_n$. 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.

KEYWORDS:

• Hecke algebras
• super-algebras
• symmetric polynomials

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W50
• 15A66
• 17A70
• 13F20

Notes

1 One may be tempted to call these supersymmetric polynomials; this term has already been coined for another notion though [20].

2 It seems more natural to put all Clifford-generators on the right, because ${\mathfrak{d}}_1$ is ${\mathfrak{C}}_I$-right linear. We shall stick to putting them on the left though in order to preserve compatibility with the notation used in [11].