2012/06/20 by Elise delMas, delMas, Elise, Tom Halverson +1 · 1 citation
Mathematics · #05E10 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #math.RT #msc:05E10
paper · pdf · doi:10.48550/arxiv.1206.4576
18 pages; Updated references; Changed section 2.2 to include the two parameter rook-Brauer algebra RB_k(x,y) defined by Mazorchuk in reference [Mz]; Added comments around equation (4.5); Accepted for publication in Communications in Algebra
openalex publication_date 2012/06/20 · arxiv created 2012/07/25 · arxiv updated 2012/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the representation theory of the rook-Brauer algebra RBk(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 RBk(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 RBk(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.