Extended nilHecke algebra and symmetric functions in type B

TL;DR

This paper introduces a type B extended nilHecke algebra, explores its action on extended polynomials, and establishes a connection to classical symmetric polynomial rings, expanding algebraic understanding in type B.

Contribution

It formulates a new type B extended nilHecke algebra and proves a Solomon-type theorem linking extended symmetric polynomials to classical symmetric functions.

Findings

Defined the type B extended nilHecke algebra.

Described its action on extended polynomials.

Proved a Solomon-type isomorphism for extended symmetric polynomials.

Abstract

We formulate a type B extended nilHecke algebra, following the type A construction of Naisse and Vaz. We describe an action of this algebra on extended polynomials and describe some results on the structure on the extended symmetric polynomials. Finally, following Appel, Egilmez, Hogancamp, and Lauda, we prove a result analogous to a classical theorem of Solomon connecting the extended symmetric polynomial ring to a ring of usual symmetric polynomials and their differentials.