Last update: 10 September 2013

It was shown in [Ram1994] that the trace of the regular representation of the Brauer algebra ${B}_{m}$ was related to the character of the irreducible representation of the Brauer algebra ${B}_{2m}$ labeled by the partition $\varnothing \text{.}$ Here we will show that this result holds in a much more general context, for any centralizer algebra.

Let $W$ be a representation of a group $G$ and let $\mathcal{Z}={\text{End}}_{G}\left(W\right)\text{.}$ Let ${W}^{*}$ be the representation of $G$ which is dual to $W$ and let ${\mathcal{Z}}^{\text{opp}}={\text{End}}_{G}\left({W}^{*}\right)\text{.}$ There is a natural identification of ${\mathcal{Z}}^{\text{opp}}$ with the algebra $\mathcal{Z}$ with the opposite multiplication. Let $\stackrel{\u02c6}{\mathcal{Z}}={\text{End}}_{G}(W\otimes {W}^{*})\text{.}$ Clearly $\mathcal{Z}\otimes {\mathcal{Z}}^{\text{opp}}\subseteq \stackrel{\u02c6}{\mathcal{Z}}\text{.}$ Let ${\mathcal{Z}}^{\varnothing}={(W\otimes {W}^{*})}^{G}$ be the $G$ invariants. ${\stackrel{\u02c6}{\mathcal{Z}}}^{\varnothing}$ is a $\stackrel{\u02c6}{\mathcal{Z}}\text{-submodule}$ of $W\otimes {W}^{*}\text{.}$ Let ${\chi}^{\varnothing}$ be the trace of $\stackrel{\u02c6}{\mathcal{Z}}$ in this representation.

Let $z,y\in \mathcal{Z}\text{.}$ Let ${y}^{*}\in {\mathcal{Z}}^{\text{opp}}$ be the element which corresponds to $y\in \mathcal{Z}\text{.}$ The bitrace of the regular representation of $\mathcal{Z}$ is given by $$\text{btr}(x,y)=\sum _{z\in Z}xzy{|}_{z},$$ where the sum is over a basis $Z$ of the algebra $\mathcal{Z}\text{.}$

$$\text{btr}(x,y)={\chi}^{\varnothing}(x\otimes {y}^{*})\text{.}$$

Proof. | |

This is a trivial consequence of the chain of isomorphisms $$\mathcal{Z}={\text{End}}_{G}\left(W\right)\simeq {(W\otimes {W}^{*})}^{G}={\stackrel{\u02c6}{\mathcal{Z}}}^{\varnothing}\text{.}$$ Let ${w}_{i}$ be a basis of $W$ and let ${w}^{i}$ be a basis of ${W}^{*}\text{.}$ Then $z\in \mathcal{Z}$ and the corresponding element ${z}^{*}\in {Z}^{\text{opp}}$ act on $W$ and ${W}^{*}$ respectively by $$z{w}_{i}=\sum _{j}{z}_{i}^{j}{w}_{j},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}z{w}^{i}=\sum _{j}{z}_{j}^{i}{w}^{j}\text{.}$$ Similarly, let $g\in G$ act on $W$ and ${W}^{*}$ by $$g{w}_{i}=\sum _{j}{g}_{i}^{j}{w}_{j},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}g{w}^{i}=\sum _{j}{\stackrel{\u02c6}{g}}_{j}^{i}{w}^{j}\text{.}$$ Then the fact that $z\in \mathcal{Z}$ and that $z\in {\mathcal{Z}}^{\text{opp}}$ implies that $$\sum {g}_{i}^{j}{z}_{j}^{k}=\sum {z}_{i}^{l}{g}_{l}^{k},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\sum {\stackrel{\u02c6}{g}}_{i}^{j}{z}_{j}^{k}=\sum {z}_{i}^{l}{\stackrel{\u02c6}{g}}_{l}^{k}\text{.}$$ Then the element ${\sum}_{i,j}{z}_{i}^{j}{w}_{j}{w}^{i}\in W\otimes {W}^{*}$ is invariant since $$g\sum _{i,j}{z}_{i}^{j}{w}_{j}{w}^{i}=\sum _{i,j,k,l}{\stackrel{\u02c6}{g}}_{l}^{i}{z}_{i}^{j}{g}_{j}^{k}{w}_{k}{w}^{l}=\sum _{i,j,k,l}{\stackrel{\u02c6}{g}}_{l}^{i}{g}_{i}^{j}{z}_{j}^{k}{w}_{k}{w}^{l}=\sum _{j,k,l}{\delta}_{l}^{j}{z}_{j}^{k}{w}_{k}{w}^{l}=\sum _{k,l}{z}_{l}^{k}{w}_{k}{w}^{l}\text{.}$$ Now let us consider the element $x\otimes {y}^{*}\in \mathcal{Z}\otimes {\mathcal{Z}}^{\text{opp}}$ acting on the invariant in ${\mathcal{Z}}^{\varnothing}={(W\otimes {W}^{*})}^{G}$ corresponding to $z\in \mathcal{Z}\text{.}$ This is the invariant in $W\otimes {W}^{*}$ given by $$(x\otimes {y}^{*})\sum _{i,j}{z}_{j}^{i}{w}_{i}{w}^{j}=\sum _{i,j,k,l}{y}_{l}^{j}{z}_{j}^{i}{x}_{i}^{k}{w}_{k}{w}^{l},$$ which clearly corresponds to the element of $\mathcal{Z}$ given by $xzy\text{.}$ $\square $ |

**Remark.** It follows that the bitrace of the regular representation of the Iwahori-Hecke algebra ${H}_{m}\left(q\right)$
of type $A$ is the same as the character of the representation of the Kosuda algebra
${H}_{m,m}\left(q\right)$
corresponding to the pair of partitions $[\varnothing ,\varnothing ]\text{.}$

This is a copy of the paper *On the trace of the regular representation of a centralizer algebra* by Arun Ram, Department of Mathematics, University of Wisconsin, Madison, WI 53706, February 14, 1994. This paper was supported in part by a National Science Foundation postdoctoral fellowship.