SOME IRREDUCIBLE REPRESENTATIONS OF BRAUER'S CENTRALIZER ALGEBRAS

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Let m, n ∈ , V be a 2m-dimensional complex vector space. The irreducible representations of the Brauer's centralizer algebra Bn(−2m) appearing in V ⊗n are in 1-1 correspondence to the set of pairs ( f ,λ ), where f ∈ with 0 ≤ f ≤ (n/2), and λ � n − 2f satisfying λ1 ≤ m. In this paper, we first show that each of these representations has a basis consists of eigenvectors for the subalgebra of Bn(−2m) generated by all the Jucys-Murphy operators, and we determine the corresponding eigenvalues. Then we identify these representations with the irreducible representations constructed from a cellular basis of Bn(−2m). Finally, an explicit description of the action of each generator of Bn(−2m) on such a basis is also given, which generalizes earlier work of (15) for Brauer's centralizer algebra Bn(m).