# Spin nilHecke algebras of classical type

## TL;DR

This paper introduces and studies spin nilHecke algebras of types B and D, revealing their structure, representations, and connections to spin symmetric polynomials, extending known results from type A.

## Contribution

It formulates new spin nilHecke algebras of types B and D, constructs faithful polynomial representations, and establishes their matrix algebra structure with spin symmetric polynomial entries.

## Key findings

- Spin nilHecke algebras are matrix algebras over spin symmetric polynomials.
- Faithful polynomial representations via odd Demazure operators are constructed.
- Results generalize known type A nilHecke algebra results to types B and D.

## Abstract

We formulate and study the spin nilHecke algebras ${}^\mathfrak{b}\!{\mathrm{NH}}_n^-$ and ${}^\mathfrak{d}\!{\mathrm{NH}}_n^-$ of type B/D, which differ from the usual nilHecke algebras by some odd signs. The type B spin nilHecke algebra is a nil version of the spin type B Hecke algebra introduced earlier by the second author and Khongsap, but not for the type D one. We construct faithful polynomial representations $\mathrm{Pol}_n^-$ of the nilHecke algebras via odd Demazure operators. We formulate the spin Schubert polynomials, and use them to show that the spin nilHecke algebras are matrix algebras with entries in a subalgebra of $\mathrm{Pol}_n^-$ consisting of spin symmetric polynomials. All these results have their counterparts for the usual nilHecke algebras over the rational field. Our work is a generalization of results of Lauda and Ellis-Khovanov-Lauda in usual/spin type A.