Modification Rules and Products of Irreducible Representations of the Unitary, Orthogonal, and Symplectic Groups

Abstract

Modification rules, expressible in terms of the removal of continuous boundary hooks, are derived which relate nonstandard irreducible representations (IR's) of the unitary, orthogonal, and symplectic groups in n dimensions to standard IR's. Tensorial methods are used to derive procedures for reducing the outer products of IR's of U(n), O(n), and Sp(n), and for reducing general IR's of U(n) specified by composite Young tableaux with respect to the subgroups O(n) and Sp(n). In these derivations the conjugacy relationship between the orthogonal and the symplectic groups is fully exploited. The results taken in conjunction with the modification rules are valid for all n.

A basis for the symplectic group branching algebra

“…Branching rules for (Sp 2n (C), Sp 2n−2 (C)), especially their combinatorial aspects, are well known (e.g., [Le71,Ki71,Ki75,KT87,KT90,Pr94]). The main goal of this paper is to investigate the branching algebra B which governs the branching of symplectic groups.…”

Stable branching rules for classical symmetric pairs

“…(a) Diagonal: The first rule 2.1.1 appears as (4.6) with (4.15) in King's paper [Ki2]. The branching rules 2.1.2 and 2.1.3 for orthogonal and symplectic groups goes back to Newell [Ne] and Littlewood [Li3].…”

“…Besides the Diagonal branching formulas, [Su] also presents a thorough treatment of the classical Littlewood restriction rules. However, in the most general form, rules 2.4.1 and 2.4.2 appear as (5.7) with (4.19), and (5.8) with (4.23) respectively in [Ki2].…”

“…Removing the stability condition for Littlewood's restriction rules is a delicate problem, which was also addressed in [EW1] and [EW2]. Classically, Newell [Ne] presents modification rules to the Littlewood restriction rules to solve the branching problem outside of the stable range (see [Su] and [Ki2]). For some recent remarks on the literature of branching rules we refer the reader to [Ki3], [Kn2], [Kn3], [Kn4] and [Pr].…”

“…Also, the late Wybourne and his students have even incorporated these results in the software package SCHUR downloadable at http://smc.vnet.net/Schur.html. This package implements all the modification rules given in [Ki2] and [BKW] that allow the stable branching rules to be generalized so as to cover all possible non-stable cases as well.…”