Character Expansions for the Orthogonal and Symplectic Groups

TL;DR

This paper derives formulas for expanding invariant functions of orthogonal and symplectic groups in terms of their characters, using a combinatorial approach, and verifies these formulas through known symmetric function expansions.

Contribution

It introduces a new combinatorial method to expand invariant functions for Sp(2N), SO(2N+1), and SO(2N) groups in terms of their characters, extending previous work on U(N).

Findings

Derived explicit expansion formulas for group functions.

Validated formulas by matching known symmetric function expansions.

Provided expansions relevant for gauge field partition functions.

Abstract

Formulas for the expansion of arbitrary invariant group functions in terms of the characters for the Sp(2N), SO(2N+1), and SO(2N) groups are derived using a combinatorial method. The method is similar to one used by Balantekin to expand group functions over the characters of the U(N) group. All three expansions have been checked for all N by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions. An expansion of the exponential of the traces of group elements, appearing in the finite-volume gauge field partition functions, is worked out for the orthogonal and symplectic groups.