Degenerate two-boundary centralizer algebras

Diagram algebras (e.g. graded braid groups, Hecke algebras, Brauer algebras) arise as tensor power centralizer algebras, algebras of commuting operators for a Lie algebra action on a tensor space. This work explores centralizers of the action of a complex reductive Lie algebra g on tensor space of the form M ⊗ N ⊗ V ⊗k. We define the degenerate two-boundary braid group Gk and show that centralizer algebras contain quotients of this algebra in a general setting. As an example, we study in detail the combinatorics of special cases corresponding to Lie algebras gln and sln and modules M and N indexed by rectangular partitions. For this setting, we define the degenerate two-boundary Hecke algebra H k as a quotient of Gk, and show that a quotient of H k is isomorphic to a large subalgebra of the centralizer. We further study the representation theory of H k 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 H k is given by combinatorial formulas.