A Brauer Algebra Theoretic Proof of Littlewood's Restriction Rules

Let U be a complex vector space endowed with an orthogonal or symplectic form, and let G be the subgroup of GL(U) of all the symmetries of this form (resp. O(U) or Sp(U)); if M is an irreducible GL(U)-module, the Littlewood's restriction rule describes M as a G-module (via restriction). In this paper we give a new representation-theoretic proof of this formula: realizing M in a tensor power U^{\otimes f} and using Schur's duality, we reduce to the problem of describing the restriction to an irreducible S_f-module of an irreducible module for the centralizer algebra of the action of G on U^{\otimes 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.