Diagram algebras (for
example, graded braid groups, Hecke algebras and 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 algebra 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 extended two-boundary Hecke algebra ℋkext as a
quotient of 𝒢k, and show that a quotient of ℋkext is isomorphic to a large
subalgebra of the centralizer. We further study the representation theory of
ℋkext 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 ℋkext is given by combinatorial
formulas.
