Lectures in Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 27 August 2013

Lecture 18

On the other hand, since z=a𝒜 χV(a)a* it follows that [z] = a𝒜 χV(a) [a*] = b a𝒜 χV(a) [a*],b b* In fact, these scalars [a*],b are equal to a*,[b], for [a*],b = h ha*h*,b = ht (ha*h*b) = ht (a*h*bh) = h a*,h*bh = h** a*,h* b(h*)* = a*,[b] (This last equality follows by the basis-independence of the [·] operator.) This now gives us a second formula for [z], namely [z]=b a𝒜χV (a)a*,[b] b* Comparing this with our previous formula for [z], and equating coefficients of b*, we obtain χVAB(b) =a𝒜χV (a)a*,[b].

Now let’s consider the special case of group algebras, with HG for H and G finite groups. We know that HG is a subalgebra. Let V be a representation of H, with V:HMd():hV(h). Let 𝔳={v1,,vd} be a basis for the H-module V, such that hvk=j=1dvjVjk(h). Let us denote VHG by VHG. Note that VHG= GHV= span{gv|gG,vV}= span{gvk|gG,vk𝔳}. Let 𝔤={g0=1,g1,,gm} be a set of (right) coset representatives for H in G, so that {H,g1H,,gmH}=G/H. Now if gG, then there is some gj𝔤 and some hH with g=gjh; one has that for vk𝔳 gvk=gjhvk =gjhvk= l=1dgj vlVlk(h). Thus the elements gjvl form a basis for VHG.

Note that G acts on G/H on the left, by permuting the cosets. For gG and gi𝔤, we can produce gj𝔤 and hH with ggi=gjh. Let π(g) be the m+1 by m+1 permutation matrix determined by the action of g on G/H. π:GMm+1() is the permutation representation of G. Now g(givk) = (ggi)vk = (gjh)vk = gj(hvk) = l=1d (gjvl) Vlk(h) g(givk) = l=1d (gjvl) Vlk (gj-1ggi) Taking traces, we find that χVHG(g) = gi𝔤 vk𝔳 g(givk) |givk = gi𝔤gi-1ggiH vk𝔳 Vkk(gi-1ggi) = gi𝔤gi-1ggiH χV(gi-1ggi) = hH1H gi𝔤gi-1ggiH χV(h-1gi-1ggih) = gih=wGw-1gwH 1HχV(wgw-1) = gGg𝒞gH 1HχV(g) (where 𝒞g is the conjugacy class of g). Our previous (general) formula for χVAB would have us believe that χVHG= a𝒜=H χV(a) a*,[b]. But for group algebras a*,[b]= { 1 ifa𝒞b 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.

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