A Brauer Algebra Theoretic Proof of Littlewoodʹs Restriction Rules

LetUbe a complex vector space endowed with an orthogonal or symplectic form, and letGbe the subgroup ofGL(U) of all the symmetrics of this form (resp.O(U) orSp(U)); ifMis an irreducibleGL(U)-module, the Littlewoodʹs restriction rule describes theG-moduleMGL(U)G. In this paper we give a new representation-theoretic proof of this formula: realizingMin a tensor powerU fand using Schurʹs duality, we reduce to the problem of describing the restriction to an irreducibleSf-module of an irreducible module for the centralizer algebra of the action ofGonU f; the latter is a quotient of the Brauer algebra, and we know the kernel of the natural epimorphism, whence we deduce the Littlewoodʹs restriction rule.