Lectures in Representation Theory

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

Last update: 27 August 2013

Lecture 17

We saw last time that the Bratteli diagram describes the branching rules for restriction from irreducible modules over B to irreducible modules over A. Note that by Schur’s Lemma and Frobenius Reciprocity, gλμ=dim ( homA (Vλ,WμAB) ) =dim ( homB (VλAB,Wμ) ) , whence that VλAB μBˆ gλμWμ, so that the Young lattice also describes branching rules for induction of irreducible A-modules.

For A semisimple, we could produce the irreducible modules Vλ by noting that Vλ=Aeiiλ, where the eijλ were the preimages of the Eij in the λ-th block within AλAˆMdλ().

Conversely, given an idempotent p in A, we can write p=pi, where the pi are minimal idempotents, and ApApi, with each Api irreducible. For each minimal idempotent pA, there exists a unique λA with ApVλ; let us indicate this association by denoting our idempotents as pλ. Now if we set Ap=V=λAˆ(Vλ)mλ, one has that p=λAˆ i=1mλ pλi and further Ap = AλAˆ i=1mλ pλi λAˆ i=1mλ Apλi

Note that ApABBp, as B-modules. (To prove this, one needs to produce a bijection BAApBp. The appropriate maps are the C-linear extensions of bapbap and bpbpp.)

Now we know that VλAB ApλAAB BpλA, for pλA a minimal idempotent in A (and naturally an idempotent of some sort in B). The branching rule (the induction version) now tells us that VλAB μBˆ gλμWμ. Thus if we express the idempotent pλA as a sum of minimal idempotents of B, we will obtain pλA=μBˆi=1gλμqμiB, where the qμiB are minimal idempotents in B with BqμiBWμ.

Let V be a representation of A, and χV the associated character, with VλAˆmλVλ and χV=λAˆmλχAλ We would like to produce a formula for χVAB. As B-modules, we know that VAB λAˆ mλVλ AB λAˆ mλμBˆ gλμWμ, and by taking traces one then obtains the formula χVAB= λAˆ μBˆ mλgλμ χBμ. This formula, while interesting, is not necessarily so nice for computational uses. With this in mind, let us define z=λAˆ mλdλA zλAA, where (as usual) the zλA are the minimal central idempotents of A, the dλA are the dimensions of the irreducible A-modules, and the mλ are the multiplicities of these irreducibles. Let {a}=𝒜 be a basis for A, and {a*}=𝒜* be its dual basis with respect to the trace of the regular representation of A. Now recall that zλA=a𝒜 dλAχAλ (a)a*, whence that z = λAˆ mλdλA zλA = λAˆ mλdλA a𝒜 dλAχAλ (a)a* = λAˆ ( a𝒜 mλχAλ (a) ) a* = λAˆ χV(a)a* We find then that the values of χV are encoded in this distinguished element z. Can we produce an element of B within which the values of χVAB are similarly encoded?

Let {b}= be a basis for B, let t=(tλ) be any nondegenerate trace for B, and let {b*}=* be the associated dual basis. Define for each xB [x]=b bxb*.

Proposition 2.60 Given the above definitions, [z]= b χVAB (b)b* .


[z] = λAˆ mλdλA [zλA] = λAˆ mλdλA i=1dλ [pλiA] = λAˆ mλdλA i=1dλ μBˆ j=1gλμ [qμjB] (The second equality is expressing zλA in terms of minimal idempotents of A, the third is expressing each of those minimal A-idempotents in terms of the minimal idempotents of B.) Note that if we take ek,kμB and symmetrize it, we have [ek,kμ] = 1i,jdνBνBˆ eijν ekkμ (eijν)* = 1i,jdνBνBˆ eijνekkμ ejiνtν = 1idνBj=kν=μ eijν ekkμ ejiν tν = i=1dν eiiνtν =1tνzνB Thus we have that [z] = λAˆ mλdλA i=1dλ μBˆ j=1gλμ [qμjB] = λAˆ mλdλA i=1dλ μBˆ j=1gλμ 1tμzμB = λAˆ μBˆ mλdλA dλAgλμ 1tμzμB = λAˆ μBˆ mλdλA gλμzμB = λAˆ μBˆ mλdλA gλμ btμ χBμ(b)b* zμB = b λAˆ μBˆ mλgλμ χBμ(b) χVAB(b) b* [z] = b χVAB (b)b*

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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