Degenerate two-boundary centralizer algebras. (Q450508)

The author studies the centralizers of the action of a complex reductive Lie algebra \(\mathfrak g\) on tensor space of the form \(M\otimes N\otimes V^{\otimes k}\). She defines the degenerate two-boundary braid algebra \(G_k\) and shows that centralizer algebras contain quotients of this algebra in a general setting. As an example, she studies in detail the combinatorics of special cases corresponding to Lie algebras \(\mathfrak{gl}_n\) and \(\mathfrak{sl}_n\) and modules \(M\) and \(N\) indexed by rectangular partitions. Furthermore, she studies the representation theory of \(\mathcal H_k^{\text{ext}}\) to find that the seminormal representations are indexed by a known family of partitions. The bases for the resulting modules are given by paths in a lattice of partitions, and the action of \(\mathcal H_k^{\text{ext}}\) is given by combinatorial formulas.