Orthogonality Relations and Supercharacter Formulas of U(m|n) Representations

TL;DR

This paper derives orthogonality relations and supercharacter formulas for the supergroup U(m|n), extending previous results to complex conjugated and mixed representations, and introduces new labeling and identities among U(N) representations.

Contribution

It provides the first orthogonality relations for U(m|n) supergroups, extending existing representation theory results and introducing a novel labeling scheme for irreducible representations.

Findings

Derived orthogonality relations for U(m|n) supergroup representations.

Obtained closed-form expressions for supercharacters and dimensions.

Proposed a new labeling method for irreducible U(m|n) representations.

Abstract

In this paper we obtain the orthogonality relations for the supergroup U(m|n), which are remarkably different from the ones for the U(N) case. We extend our results for ordinary representations, obtained some time ago, to the case of complex conjugated and mixed representations. Our results are expressed in terms of the Young tableaux notation for irreducible representations. We use the supersymmetric Harish-Chandra-Itzykson-Zuber integral and the character expansion technique as mathematical tools for deriving these relations. As a byproduct we also obtain closed expressions for the supercharacters and dimensions of some particular irreducible U(m|n) representations. A new way of labeling the U(m|n) irreducible representations in terms of m + n numbers is proposed. Finally, as a corollary of our results, new identities among the dimensions of the irreducible representations of the unitary group U(N) are presented.