## On the trace of the regular representation of a centralizer algebra

Last update: 10 September 2013

## Trace of the regular representation

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

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

$btr(x,y)= χ∅ (x⊗y*).$

 Proof. This is a trivial consequence of the chain of isomorphisms $𝒵=EndG(W)≃ (W⊗W*)G= 𝒵ˆ∅.$ Let ${w}_{i}$ be a basis of $W$ and let ${w}^{i}$ be a basis of ${W}^{*}\text{.}$ Then $z\in 𝒵$ and the corresponding element ${z}^{*}\in {Z}^{\text{opp}}$ act on $W$ and ${W}^{*}$ respectively by $zwi=∑j zijwj, andzwi= ∑jzjiwj.$ Similarly, let $g\in G$ act on $W$ and ${W}^{*}$ by $gwi=∑jgij wj,andgwi =∑jgˆji wj.$ Then the fact that $z\in 𝒵$ and that $z\in {𝒵}^{\text{opp}}$ implies that $∑gijzjk= ∑zilglk, and ∑gˆijzjk =∑zilgˆlk.$ Then the element ${\sum }_{i,j}{z}_{i}^{j}{w}_{j}{w}^{i}\in W\otimes {W}^{*}$ is invariant since $g∑i,jzij wjwi= ∑i,j,k,l gˆlizij gjkwkwl= ∑i,j,k,l gˆligij zjkwkwl= ∑j,k,l δljzjkwk wl=∑k,l zlkwkwl.$ Now let us consider the element $x\otimes {y}^{*}\in 𝒵\otimes {𝒵}^{\text{opp}}$ acting on the invariant in ${𝒵}^{\varnothing }={\left(W\otimes {W}^{*}\right)}^{G}$ corresponding to $z\in 𝒵\text{.}$ This is the invariant in $W\otimes {W}^{*}$ given by $(x⊗y*) ∑i,jzji wiwj= ∑i,j,k,lylj zjixikwk wl,$ which clearly corresponds to the element of $𝒵$ 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 $\left[\varnothing ,\varnothing \right]\text{.}$

## Notes and References

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.