On immanant functions related to Weyl groups of $A_n$

TL;DR

This paper explores the generalization of matrix immanants to develop new symmetric and antisymmetric orbit functions associated with Weyl groups of type A_n, including their properties and orthogonality relations.

Contribution

It introduces new orbit functions derived from immanants related to Weyl groups of A_n, expanding the understanding of their properties and orthogonality.

Findings

New symmetric and antisymmetric orbit functions are defined.

Orthogonality relations for these functions are established.

Several illustrative examples are provided.

Abstract

In this work we recall the definition of matrix immanants, a generalization of the determinant and permanent of a matrix. We use them to generalize families of symmetric and antisymmetric orbit functions related to Weyl groups of the simple Lie algebras $A_n$. The new functions and their properties are described, in particular we give their continuous orthogonality relations. Several examples are shown.