A basis for the diagonally signed-symmetric polynomials

TL;DR

This paper constructs an explicit free basis for the ring of diagonally signed-symmetric polynomials under the hyperoctahedral group action, using signed descent monomials, advancing understanding of invariant polynomial structures.

Contribution

It provides a new explicit basis for the invariant ring of diagonally signed-symmetric polynomials as a module over symmetric polynomials, using signed descent monomials.

Findings

Explicit free basis constructed for the invariant ring.

Basis expressed as a module over symmetric polynomials.

Utilizes signed descent monomials for basis construction.

Abstract

Let n>0 be an integer and let B_{n} denote the hyperoctahedral group of rank n. The group B_{n} acts on the polynomial ring Q[x_{1},...,x_{n},y_{1},...,y_{n}] by signed permutations simultaneously on both of the sets of variables x_{1},...,x_{n} and y_{1},...,y_{n}. The invariant ring M^{B_{n}}:=Q[x_{1},...,x_{n},y_{1},...,y_{n}]^{B_{n}} is the ring of diagonally signed-symmetric polynomials. In this article we provide an explicit free basis of M^{B_{n}} as a module over the ring of symmetric polynomials on both of the sets of variables x_{1}^{2},..., x^{2}_{n} and y_{1}^{2},..., y^{2}_{n} using signed descent monomials.