The Rook-Brauer Algebra

Authors

Elise G. delMas, Macalester CollegeFollow

Document Type

Honors Project - Open Access

Abstract

We introduce an associative algebra RB_k(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 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. 26.
https://digitalcommons.macalester.edu/mathcs_honors/26