The centralizer algebra of the diagonal action of the group GLn(ℂ) in a mixed tensor space

We consider the walled Brauer algebra Br _{k, l}(n) introduced by V. Turaev and K. Koike. We prove that it is a subalgebra of the Brauer algebra and that it is isomorphic, for sufficiently large n ∈ ℕ, to the centralizer algebra of the diagonal action of the group GL_n(ℂ) in a mixed tensor space. We also give the presentation of the algebra Br _{k, l}(n) by generators and relations. For a generic value of the parameter, the algebra is semisimple, and in this case we describe the Bratteli diagram for this family of algebras and give realizations for the irreducible representations. We also give a new, more natural proof of the formulas for the characters of the walled Brauer algebras. Bibliography: 29 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 170–198.