Degenerate two-boundary centralizer algebras

TL;DR

This paper introduces and studies degenerate two-boundary centralizer algebras related to Lie algebra actions on tensor spaces, providing new algebraic structures, their quotients, and combinatorial representation theory insights.

Contribution

It defines the degenerate two-boundary braid algebra and its quotient, the extended two-boundary Hecke algebra, and analyzes their representation theory in the context of Lie algebra tensor actions.

Findings

Centralizer algebras contain quotients of the degenerate two-boundary braid algebra.

The extended two-boundary Hecke algebra's quotient is isomorphic to a large subalgebra of the centralizer.

Seminormal representations are indexed by partitions, with bases given by paths in a lattice of partitions.

Abstract

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 $\mathfrak{g}$ on tensor space of the form $M \otimes N \otimes V^{\otimes k}$. We define the degenerate two-boundary braid algebra $\mathcal{G}_k$ 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 $\mathfrak{gl}_n$ and $\mathfrak{sl}_n$ and modules $M$ and $N$ indexed by rectangular partitions. For this setting, we define the degenerate extended two-boundary Hecke algebra $\mathcal{H}_k^{\mathrm{ext}}$ as a quotient of $\mathcal{G}_k$, and show that a quotient of $\mathcal{H}_k^{\mathrm{ext}}$ is isomorphic to a large subalgebra of the centralizer. We further study the representation theory of $\mathcal{H}_k^{\mathrm{ext}}$ 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 $\mathcal{H}_k^{\mathrm{ext}}$ is given by combinatorial formulas.