Some irreducible representations of Brauer's centralizer algebras

Let m, n ∈ ℕ, V be a 2m-dimensional complex vector space. The irreducible representations of the Brauer's centralizer algebra B _n(-2m) appearing in V^⊗n are in 1-1 correspondence to the set of pairs (f, λ), where f ∈ ℤ 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 B_n(-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 B_n(-2m). Finally, an explicit description of the action of each generator of B_n(-2m) on such a basis is also given, which generalizes earlier work of [15] for Brauer's centralizer algebra B_n(m).