Lectures in Representation Theory

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

Last update: 27 August 2013

Lecture 21

The following theorem due to Burnside can be found in Burnside, Proc. London Math. Soc. (2), 3, 1905 p. 430. In his ”Classical Groups”, Weyl also directs the reader to Frobenius, Schur, Siztungsber Preuss. Akad. 1906, p. 209.

Theorem 3.18 (Burnside) Let A be an arbitrary algebra and let W:AMd() be an irreducible representation of A. Then W(A)=Md().


Without loss, W is faithful, since otherwise we may consider the representation W:AkerWMd(), which has the same image as A.

Choose a nonzero aA. By faithfulness of W, we may choose wW such that aw0. Since W is an irreducible A-module, Aaw=W. Thus, there exists a bA such that baw=w, from which it follows that (ba)kw=w for all k. In particular, ba is not nilpotent, so a is not in A, the largest nil ideal. But a was an arbitrary nonzero element, so A=0, hence A is semisimple.

If A is not simple, we may write A=IJ as a direct sum of two-sided ideals. Since W is faithful, we may choose w0 such that Iw0 Then Iw is a submodule of W and so irreducibility implies Iw=W. But JW=JIw=0, which contradicts the faithfulness of W. Thus A must be simple, so we must have AMn(). By 1.23, A has a unique irreducible module, namely W, so n=d.

Last time (3.16) we showed that given an arbitrary algebra A and a completely decomposable (finite dimensional) representation V, whose decomposition is given by V=λVˆmλVλ, then the algebra of the representation is given by V(A)=λVˆ Imλ(Vλ(A)). From the above theorem, writing dλ=dimVλ, we have V(A)=λVˆ Imλ(Md()). From proposition 3.17, recall that V(A) λVˆ Mmλ(Idλ()).

Suppose B is an arbitrary algebra which acts on V such that V(B)=V(A). Let B={bB|bV=0} be the kernel of the representation of B on V. Then we have a representation V: BB Md() b+B V(b) which is, by definition, injective. Let C=BB; by hypothesis, the map V:CV(A) is an isomorphism.

Hence, by proposition 3.17, C is semisimple, and the irreducible representations of C are indexed by λVˆ, and have dimensions mλ. Since V(c)V(A), by the above decomposition V(c)= ( c1Id1 c2Id2 c|Vˆ|Id|Vˆ| ) where cλ=(ci,jλ) are mλ×mλ complex matrices.

Define maps Cλ:CMmλ() by Cλ(c)=cλ. These are the irreducible representations of C, but they extend to irreducible representations of B via the composition BBB=CCλMmλ()

Suppose V and W are vector spaces with bases {vi}1im and {wj}1jn. Then VW is a vector space with basis {vivj|1im,1jn}.

If A and B any are algebras, then AB is a vector space which becomes an algebra under the multiplication (a1b1) (a2b2)= (a1a2b1b2).

If V and W are modules for A and B, respectively, then VW is an AB module with the action defined by (ab) (vw) =avbw

Problem 3.19 Check that the multiplication in AB is well-defined and that AB satisfies the axiom of an algebra over . Check that the action of AB on VW is a well-defined module action.

If V(a)=(ai,j)1i,jm and W(b)=(bk,)1k,n with respect to bases {vi} and {wk} of V and W, respectively, then (VW)(ab) vjw= i=1m ai,jvi j=1n bk,wk= i,kai,j bk,vi wk. If we order the basis {viwj} lexicographically (1,1), , (1,n), (2,1), , then the (i,j)(k,) entry of the matrix of (VW)(ab) is ai,jbk,.

On the other hand, the matrix V(a)W(b) is also a mn×mn matrix, so acts on VW. If we consider the block structure given by V(a)W(b)=i,j=1mai,jEi,jW(b), and label the rows and columns in the form (x,y) where x indicates the number of the n×n block row or block column and y indicates the row or column within that block, then the (i,j)×(k,) entry is the (k,) entry of ai,jW(b), namely ai,jbk,.

Proposition 3.20 If A and B are algebras with representations V and W, respectively, then VW is an AB-module, and the matrix of the representation is given by (VW) (ab) = V(a) W(b).

Apply these tensor product constructions to the previous situation where A is an arbitrary algebra with completely reducible module VλmλVλ, and where B is an arbitrary algebra with V(B)=V(A).

The Vλ represent a subset of the isomorphism classes of the irreducible A-modules. The representations Cλ:BMmλ() are, similarly, a subset of the irreducible representations of B.

The algebras A and B both act on V. Define an action of BA on V by (ba)v=bav. This is well-defined on B×A since it is the linearization of a bilinear map on B×A. Moreover, since A and B commute on V, we have (b1a1) (b2a2)v= b1a1b2a2v= b1b2a1a2v= (v1v2a1a2)v. Hence, this represents a well-defined module action. This brings us to the most important theorem of the course.

Theorem 3.21 Let A, B and V be as above. Then as BA representations, VλVˆ CλVλ where Cλ are irreducible representations of B and Vλ are irreducible representations of A.


Let n=|Vˆ|. From the decompositions V(A)=λImλVλ and V(A)λMmλ(Idλ()) we have shown that the representations of A and B on V decompose into irreducible representations as V(a) = ( Im1V1(a) ImnVn(a) ) V(b) = ( C1(b)Id1 Cn(b)Idn ) Note that each of the matrices above are block diagonal matrices, with block sizes of mλdλ×mλdλ, as λ runs over Vˆ. Thus, the BA action on V defined by (ab)v=bav is represented by the matrices of the form V(ba)=V(b)V(a)= ( C1(b)V1(a) C2(b)V2(a) Cn(b)Vn(a) ) . Thus the BA representation decomposes as V=λVˆCλVλ.

Remarks. Note that V(A) has the same center as V(A), since both are block diagonal with the same block sizes.

Note also that if we suppose A and B are semisimple, and that Vλ,λAˆ, and Vμ,μBˆ are representatives of the isomorphism classes of irreducible A and B modules, respectively, then AB is semisimple and the set of all VλVμ, λAˆ, μBˆ, are a complete set of irreducible modules for AB.

However, even if V(A) is semisimple (as always, V(A) is semisimple), the decomposition of V into V(A)V(A) modules contains only those irreducibles of the form CλVλ. There are no summands isomorphic to the irreducible V(A)V(A) modules of the form CμVλ for λμ.

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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