Walled Signed Brauer Algebras as Centralizer Algebras of the Mixed Tensor Representation of Lie Superalgebras

Abstract

In this paper, we prove that the walled signed Brauer algebra − → D r,s (x) is embedded in a canonical fashion as a subalgebra of a walled Brauer algebra B 2r,2s (x) and the walled signed Brauer algebra − → D r,s (x) is the centralizer algebra of σ = (1 2)(3 4) . . . (2r−1 2r)(2r+1 2r+2) . . . (2r+2s−1 2r+2s) in the walled Brauer algebra B 2r,2s (x). Finally we prove that, if 2r+2s < (m+1)(n+1), the walled signed Brauer algebra − → D r,s (δ) is isomorphic to the centralizer algebra End g 2r+2s (V ⊗r ⊗ (V * ) ⊗s ) op of the g 2r+2s -action on the mixed tensor space (V ⊗r ⊗ (V * ) ⊗s ); where δ = m − n, g = gl(m, n) is the complex general linear Lie superalgebra, V = W ⊗ W, W = C m|n is the natural representation of g and V * is the dual of V.