## Lectures in Representation Theory

Last update: 27 August 2013

## Lecture 18

On the other hand, since $z=∑a∈𝒜 χV(a)a*$ it follows that $[z] = ∑a∈𝒜 χV(a) [a*] = ∑b∈ℬ ∑a∈𝒜 χV(a) ⟨[a*],b⟩ b*$ In fact, these scalars $⟨\left[{a}^{*}\right],b⟩$ are equal to $⟨{a}^{*},\left[b\right]⟩,$ for $⟨[a*],b⟩ = ∑h∈ℬ ⟨ha*h*,b⟩ = ∑h∈ℬt→ (ha*h*b) = ∑h∈ℬt→ (a*h*bh) = ∑h∈ℬ ⟨a*,h*bh⟩ = ∑h*∈ℬ* ⟨ a*,h* b(h*)* ⟩ = ⟨a*,[b]⟩$ (This last equality follows by the basis-independence of the $\left[·\right]$ operator.) This now gives us a second formula for $\left[z\right],$ namely $[z]=∑b∈ℬ ∑a∈𝒜χV (a)⟨a*,[b]⟩ b*$ Comparing this with our previous formula for $\left[z\right],$ and equating coefficients of ${b}^{*},$ we obtain $χV↑AB(b) =∑a∈𝒜χV (a)⟨a*,[b]⟩.$

Now let’s consider the special case of group algebras, with $H\subseteq G$ for $H$ and $G$ finite groups. We know that $ℂH\subseteq ℂG$ is a subalgebra. Let $V$ be a representation of $H,$ with $V:ℂH\to {M}_{d}\left(ℂ\right):h↦V\left(h\right)\text{.}$ Let $𝔳=\left\{{v}_{1},\dots ,{v}_{d}\right\}$ be a basis for the $ℂH\text{-module}$ $V,$ such that $h{v}_{k}={\sum }_{j=1}^{d}{v}_{j}{V}_{jk}\left(h\right)\text{.}$ Let us denote $V{↑}_{ℂH}^{ℂG}$ by $V{↑}_{H}^{G}\text{.}$ Note that $V{↑}_{H}^{G}=ℂG{\otimes }_{ℂH}V=\text{span} \left\{g\otimes v | g\in G,v\in V\right\}=\text{span} \left\{g\otimes {v}_{k} | g\in G,{v}_{k}\in 𝔳\right\}\text{.}$ Let $𝔤=\left\{{g}_{0}=1,{g}_{1},\dots ,{g}_{m}\right\}$ be a set of (right) coset representatives for $H$ in $G,$ so that $\left\{H,{g}_{1}H,\dots ,{g}_{m}H\right\}=G/H\text{.}$ Now if $g\in G,$ then there is some ${g}_{j}\in 𝔤$ and some $h\in H$ with $g={g}_{j}h\text{;}$ one has that for ${v}_{k}\in 𝔳$ $g⊗vk=gjh⊗vk =gj⊗hvk= ∑l=1dgj ⊗vlVlk(h).$ Thus the elements ${g}_{j}\otimes {v}_{l}$ form a basis for $V{↑}_{H}^{G}\text{.}$

Note that $G$ acts on $G/H$ on the left, by permuting the cosets. For $g\in G$ and ${g}_{i}\in 𝔤,$ we can produce ${g}_{j}\in 𝔤$ and $h\in H$ with $ggi=gjh.$ Let $\pi \left(g\right)$ be the $m+1$ by $m+1$ permutation matrix determined by the action of $g$ on $G/H\text{.}$ $\pi :G\to {M}_{m+1}\left(ℂ\right)$ is the permutation representation of $G\text{.}$ Now $g(gi⊗vk) = (ggi)⊗vk = (gjh)⊗vk = gj⊗(hvk) = ∑l=1d (gj⊗vl) Vlk(h) g(gi⊗vk) = ∑l=1d (gj⊗vl) Vlk (gj-1ggi)$ Taking traces, we find that $χV↑HG(g) = ∑gi∈𝔤 ∑vk∈𝔳 g(gi⊗vk) |gi⊗vk = ∑gi∈𝔤gi-1ggi∈H ∑vk∈𝔳 Vkk(gi-1ggi) = ∑gi∈𝔤gi-1ggi∈H χV(gi-1ggi) = ∑h∈H1H ∑gi∈𝔤gi-1ggi∈H χV(h-1gi-1ggih) = ∑gih=w∈Gw-1gw∈H 1HχV(wgw-1) = ∑g′∈Gg′∈𝒞g∩H 1HχV(g′)$ (where ${𝒞}_{g}$ is the conjugacy class of $g\text{).}$ Our previous (general) formula for ${\chi }_{V{↑}_{A}^{B}}$ would have us believe that $χV↑HG= ∑a∈𝒜=H χV(a) ⟨a*,[b]⟩.$ But for group algebras $⟨a*,[b]⟩= { 1 if a∈𝒞b 0 otherwise$ so our two formulae are in agreement.

## Notes and References

This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.