#### Title

#### Document Type

Honors Project

#### Abstract

We introduce an associative algebra RB_{k}(x) that has a basis of rook-Brauer diagrams. These diagrams correspond to partial matchings on 2*k* 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 RB_{k}(x) is generated by special diagrams s_{i}, t_{i} (1 <= i < *k*) and p_{j} (1 <= j <= *k*), where the s_{i} are the simple transpositions that generated the symmetric group S_{k}, the t_{i} are the "contraction maps" which generate the Brauer algebra B_{k}(x), the p_{j} are the "projection maps" that generate the rook monoid R_{k}. We prove that for a positive integer *n*, the algebra RB_{k}(*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.

#### Recommended Citation

delMas, Elise G., "The Rook-Brauer Algebra" (2012). *Mathematics, Statistics, and Computer Science Honors Projects.* Paper 26.

http://digitalcommons.macalester.edu/mathcs_honors/26

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