# Generating functions for some series of characters of classical Lie groups

## TL;DR

This paper extends and generalizes generating functions for series of characters of classical Lie groups, providing explicit recurrence relations and tabulated coefficients, enriching the understanding of symplectic and orthogonal group representations.

## Contribution

It introduces new explicit recurrence relations for expansion coefficients of characters across all classical Lie groups, unifying previous identities and exploring dual pair approaches.

## Key findings

- Derived explicit recurrence relations for expansion coefficients.
- Tabulated coefficients as functions of highest weight parameters.
- Showed expansions in spin orthogonal characters can be derived from non-spin cases.

## Abstract

There exist a number of well known multiplicative generating functions for series of Schur functions. Amongst these are some related to the dual Cauchy identity whose expansion coefficients are rather simple, and in some cases periodic in parameters specifying the Schur functions. More recently similar identities have been found involving expansions in terms of characters of the symplectic group. Here these results are extended and generalised to all classical Lie groups. This is done through the derivation of explicit recurrence relations for the expansion coefficients based on the action of the Weyl groups of both the symplectic and orthogonal groups. Copious results are tabulated in the form of explicit values of the expansion coefficients as functions of highest weight parameters. An alternative approach is then based on dual pairs of symplectic and/or orthogonal groups. A byproduct of this approach is that expansions in terms of spin orthogonal group characters can always be recovered from non-spin cases.