SOME IRREDUCIBLE REPRESENTATIONS OF BRAUER'S CENTRALIZER ALGEBRAS

Abstract

Let $m, n\in\Bbb N$, $V$ be a $2m$-dimensional complex vector space. The irreducible representations of the Brauer's centralizer algebra $B_n(-2m)$ appearing in $V^{\otimes{n}}$ are in 1–1 correspondence to the set of pairs $(\,f,\lamda)$, where $f\in\Z$ with $0\leq f\leq [n/2]$, $and$ $\lam\vdash n-2f$ satisfying $\lam_1\leq m$. In this paper, we first show that each of these representations has a basis consists of eigenvectors for the subalgebra of $B_n(-2m)$ generated by all the Jucys-Murphy operators, and we determine the corresponding eigenvalues. Then we identify these representations with the irreducible representations constructed from a cellular basis of $B_n(-2m)$. Finally, an explicit description of the action of each generator of $B_n(-2m)$ on such a basis is also given, which generalizes earlier work of [15] for Brauer's centralizer algebra $B_n(m)$.