Representations of the Rook-Brauer Algebra

TL;DR

This paper explores the representation theory of the rook-Brauer algebra, establishing its Schur-Weyl duality with the orthogonal group and constructing explicit irreducible representations.

Contribution

It proves the Schur-Weyl duality between the rook-Brauer algebra and the orthogonal group and constructs explicit irreducible representations using Bratteli diagrams.

Findings

RB_k(n) is the centralizer algebra of O(n) on tensor space

Explicit irreducible representations are constructed for semisimple cases

Rook-Brauer algebra generalizes several classical diagram algebras

Abstract

We study the representation theory of the rook-Brauer algebra RB_k(x), also called the partial Brauer algebra. This algebra has a basis of "rook-Brauer" diagrams, which are Brauer diagrams that allow for the possibility of missing edges. The Brauer, Temperley-Lieb, Motzkin, rook monoid, and symmetric group algebras are all subalgebras of the rook-Brauer algebra. We prove that RB_k(n) is the centralizer algebra of the complex orthogonal group O(n) acting on the k-fold tensor power of the sum of its 1-dimensional trivial module and its n-dimensional defining module, and thus the rook-Brauer algebra and the orthogonal group are in Schur-Weyl duality on this tensor space. In the case where the parameter x is chosen so that RB_k(x) is semisimple, we use its Bratteli diagram to explicitly construct a complete set of irreducible representations for the rook-Brauer algebra as the span of paths in this diagram. These are analogs of Young's seminormal representations of the symmetric group.