Tensor product representations of general linear groups and their connections with Brauer algebras (Q1330071)

Tensor product representations of the general linear group \(G=\text{GL}(r,\mathbb{C})\) are studied in detail in the framework of the known Schur-Weyl duality concept linking the representation theories of general linear and symmetric groups. The group \(\text{GL}(r,\mathbb{C})\) is considered as the group of \(r \times r\) invertible complex matrices which act on the space \(V=\mathbb{C}^ r\) of \(r \times 1\) complex matrices by matrix multiplication making \(V\) into a \(G\)-module. The tensor product \(T = (\bigotimes^ p V) \otimes (\bigotimes^ q V^*)\) of \(p\) copies of \(V=\mathbb{C}^ r\) and \(q\) copies of the dual \(G\)-module \(V^*\) of \(V\) is investigated. Using the insertion schemes (Stembridge, 1987, 1989) the maximal vectors of \(T\) relative to the Lie algebra \({\mathfrak g} = \text{gl}(r,\mathbb{C})\) of \(G\) are found and the explicit decomposition of \(T\) into irreducible \(\mathfrak g\)-summands (and hence \(G\)-summands) by a method similar to the BBL approach (Benkart, Britten, Lemire, 1990) is obtained. The resulting maximal vectors of \(T\) of a given weight allow to constitute a module for the centralizer algebra \({\mathcal C} = \text{End}_{{\mathfrak g}}(T)\) of all transformations commuting with \(G\) and hence also the \(\mathcal A\)-module for the algebra \(\mathcal A\) of endomorphisms of \(T\) generated by the symmetric group \(S_ P \otimes S_ Q\) where \(P = (1,\dots, p)\) and \(Q = (p+1,\dots, p+q)\). The irreducible representations of the algebra \(\mathcal C\) are constructed. Finally, the centralizer algebra \(\mathcal C\) of the Lie algebra \(\mathfrak g\) on \(T = (\bigotimes^ p V) \otimes (\bigotimes^ q V^*)\) is identified with a certain subalgebra \(B^{(2)}_{p,q}\) of the Brauer algebra \(B^{(2)}_{p + q}\), which plays the role of the group algebra of the symmetric group in the Schur-Weyl duality.