A generalization of the brauer algebra

Details A generalization of the brauer algebra A generalization of the brauer algebra

2009-06-18

arxiv.org/abs/0906.3428v1

Mathematics

Combinatorics

26 pages

Scientific paper

We study two variations of the Brauer algebra $B_n(x)$. The first is the algebra $A_n(x)$, which generalizes the Brauer algebra by considering loops. The second is the algebra $L_n(x)$, the $A_n(x)$-subalgebra generated by diagrams without horizontal arcs. $A_n(x)$ and $L_n(x)$ have for $x \neq 0$ an hereditary-chain indexed by all integers. Following the ideas of Martin in the context of the partition algebra, and Doran et al. for the Brauer algebra, we study semisimplicity of $A_n(x)$ using restriction and induction in $A_n(x)$ and $L_n(x)$. Our main result is that $A_n(x)$ is semisimple if $x \notin Z$ and that $L_n(x)$ is semisimple if $x \neq 0$.

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