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A Brauer Algebra Theoretic Proof of Littlewood's Restriction Rules

✍ Scribed by Fabio Gavarini

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 278 KB
Volume: 212
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1998.7536

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✦ Synopsis

Let U be a complex vector space endowed with an orthogonal or symplectic Ž . Ž form, and let G be the subgroup of GL U of all the symmetrics of this form resp.

. In this paper we give a new representation-G theoretic proof of this formula: realizing M in a tensor power U m f and using Schur's duality, we reduce to the problem of describing the restriction to an irreducible S -module of an irreducible module for the centralizer algebra of the f action of G on U m f ; the latter is a quotient of the Brauer algebra, and we know the kernel of the natural epimorphism, whence we deduce the Littlewood's restriction rule.

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✍ Fabio Gavarini; Paolo Papi 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 304 KB

Let G be either Sp V or O V . Using an action of the Brauer algebra, we k Ž mm . mm describe the subspace T V :V of tensors of valence k as an induced representation of the symmetric group S . As an application, we recover a special m case of Littlewood's restriction rule, affording the decompositio