We introduce an associative algebra RBk(x) that has a basis of rook-Brauer diagrams. These diagrams correspond to partial matchings on 2k vertices. The rook-Brauer algebra contains the group algebra of the symmetric group, the Brauer algebra, and the rook monoid algebra as subalgebras. We show that the basis of RBk(x) is generated by special diagrams si, ti (1 <= i < k) and pj (1 <= j <= k), where the si are the simple transpositions that generated the symmetric group Sk, the ti are the "contraction maps" which generate the Brauer algebra Bk(x), the pj are the "projection maps" that generate the rook monoid Rk. We prove that for a positive integer n, the algebra RBk(n+1) is the centralizer algebra of the orthogonal group O(n) acting on the k-fold tensor power of the sum of its 1-dimensional trivial module and n-dimensional defining module.
delMas, Elise G., "The Rook-Brauer Algebra" (2012). Mathematics, Statistics, and Computer Science Honors Projects. Paper 26.
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