Degenerate two-boundary centralizer algebras

Abstract

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 twoboundary braid algebra Ᏻ k 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 gl n and sl n and modules M and N indexed by rectangular partitions. For this setting, we define the degenerate extended two-boundary Hecke algebra Ᏼ ext k as a quotient of Ᏻ k , and show that a quotient of Ᏼ ext k is isomorphic to a large subalgebra of the centralizer. We further study the representation theory of Ᏼ ext 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 Ᏼ ext k is given by combinatorial formulas.

A physical approach to the classification of indecomposable Virasoro representations from the blob algebra

“…We will see below that the (full) centralizer of U q sl 2 is actually exhausted by B res (2N, n, y) even at root of unity cases, this result for generic cases was also discussed in [53]. For recent work in the mathematics literature dealing with the blob algebra as a centralizer, see [64,65].…”

“…We will come back to this when we study the decomposition in non-generic cases. We finally note that the decomposition (65) shows that the boundary XXZ spin-chain is a faithful representation of B res (2N, n, y).…”

Two Boundary Centralizer Algebras for $\mathfrak{gl}(n|m)$

“…Inspired by this result and related literature, Daugherty [4] studied the two boundary case, where the underlying space is M ⊗ N ⊗ V ⊗d . The modules M and N are taken to be highest weight representations of the Lie algebra gl n whose corresponding Young diagrams are rectangles.…”

Representations of Degenerate Affine Hecke Algebra of Type $C_n$ Under the Etingof-Freund-Ma Functor

“…In [Ram03], Ram introduced the chambers and local regions and described the representations of the affine Hecke algebra. In [Dau12], Daugherty introduce the combinatorial description of representations of degenerate extended two-boundary Hecke algebra. In [DR18], Daugherty and Ram gave a Schur-Weyl duality approach to the affine Hecke algebra of type C n .…”

“…In [DR18], Daugherty and Ram gave a Schur-Weyl duality approach to the affine Hecke algebra of type C n . This paper focuses on the representations of the degenerate affine Hecke algebra of type C n and gives a combinatorial description which is similar to the combinatorial description in [Dau12] and [DR18] but is via a different structure, the Etingof-Freund-Ma functor. This paper is arranged as follow: Section 2-4 are about the Etingof-Freund-Ma functor, the degenerate affine Hecke algebra of type C n and GL Nmodules.…”