Last update: 6 November 2012
The trace, of a matrix is the sum of its diagonal elements, A trace on an algebra is a map such that for all
Every representation of determines a trace on given by where A trace is nondegenerate if for each there exists such that A trace on determines a symmetric bilinear form on given by
Suppose is finite dimensional and let be a basis of A basis of is dual to with respect to the form if
The Gram matrix of is the matrix
Suppose that exists and that an matrix such that
In matrix notation this says that So must be Conversely, if then defining by (2.4) determines a dual basis This shows that exists if and only if is invertible and that if it exists it is unique.
(2.5) Proposition. If is a trace on a finite dimensional algebra with basis and is given by (2.2) then the Gram matrix is invertible if and only if is nondegenerate.
The trace is degenerate if and only if there exists such that for all This is the same as saying that for each basis element If we have that the satisfy the system of equations
This system has a nontrivial solution if and only if the matrix is singular.
Let be a finite dimensional algebra with a nondegenerate trace and let be a basis of Let be the dual basis to with respect to the form given by (2.2). For let denote the element of such that Let and be representations of of dimensions and respectively.
(2.6) Proposition. Let be any matrix with entries in If
then, for any
If and are irreducible then Schur's lemma gives that if and are inequivalent and that if then for some
Let be a finite dimensional algebra. The action of on itself by multiplication on the left turns into an The resulting representation is the regular representation of and we denote it by The set is the same as the set but we distinguish elements of by writing As usual we denote the algebra of this representation by We denote the trace of this representation by tr. Notice that the trace tr of the regular representation can be given by
(2.8) Theorem. If is a finite dimensional algebra such that the regular representation has nondegenerate trace then every representation of is completely decomposable.
Let tr denote the trace of the regular representation. Let be a basis of and for each let denote the element of the dual basis to with respect to the trace tr such that
Let be a representation of of dimension and let be an irreducible invariant subspace of Let be an arbitrary projection of onto Define
Then, by (2.6), we know that
Since is an subspace, Since is irreducible is either 0 or
Let If then
This shows that for all Since tr is nondegenerate we have that
Now let Then since we have
Let and let Notice that for all So is an subspace of Since, for every we have If then
If is irreducible then we are done. If not apply the same process again with in place of Since is finite dimensional continuing this process will eventually produce a decomposition of into irreducible representations.
Now let be a finite dimensional algebra such that the trace tr of the regular representation of is nondegenerate. Let be a basis of and for each let denote the element of the dual basis to with respect to the trace tr such that Let be a faithful representation of By (2.8) we know that can be completely decomposed into irreducible representations. Choose a maximal set of nonisomorphic irreducible representations appearing in the decomposition of Let and define We view as an algebra of block diagonal matrices with one block for each is a subalgebra of in a natural way. Let denote the matrix with 1 in the entry of the block and 0 everywhere else and let be the matrix which is the identity on the block and 0 everywhere else.
For each let denote the entry of the matrix Then
So the row of is all zeros except for in the spot and all other rows of are zero. So
for some We can determine by setting to get
Since the trace of the regular representation was used to construct the we have, (2.9), that giving
So and we can write (2.11) as
Since we have expressed each as a linear combination of basis elements of we have that for every and But the form a basis of So Then We have proved the following theorem.
(2.12) Theorem. (Artin-Wedderburn) If is a finite dimensional algebra such that the trace of the regular representation of is nondegenerate, then, for some set of positive integers
Choose a basis of ( defined in Ex 3) and extend this basis to a basis of The Gram matrix with respect to this basis is of the form where denotes the Gram matrix on So the rank of the Gram matrix is certainly less than or equal to .
Suppose that the rows of are linearly dependent. Then for some contants not all zero, for all So This implies that This is a contradiction to the construction of the So the rows of are linearly independent.
Thus the rank of the Gram matrix is or equivalently the corank of the Gram matrix of is equal to the dimension of the radical Thus the trace of the regular representation of is nondegenerate iff
We show that is nondegenerate on ie that if , then there exists such that Since is a nonzero matrix there exists some such that Thus So there exists some such that Now is an -invariant subspace of and not 0 since Thus . So there exists some such that This shows that is not nilpotent. So So is nondegenerate on This means that for some But since by Schur's lemma where we see that
A linear transformation of is in the centraliser of if for every element and Let Then So acts on by right multiplication on Conversely it is easy to see that the action of right multiplication commutes with the action of left mutliplication since for all and So the centraliser algebra of the regular represnetation is the algebra of matrices determined by the action of right multiplication of elements of
The approach to the theory of semisimple algebras that is presented in this section and the following section follows closely a classical approach to the representation theory of finite groups, see for example [Ser1977] or [Ham1962]. Once one has the analogue of the symmetrization process for finite groups, the only nontrivial step in the theory that is not exactly analogous to the theory for finite groups is formula (2.9).
I discovered this method after reading the sections of [CRl] concerning Frobenius and symmetric algebras. Frobenius and symmetric algebras were introduced by R. Brauer and C. Nesbitt, [BNe1937] and [Nes1938]. T. Nakayama [Nak1941] has a version of Theorem (2.6) and R. Brauer [Bra1945] proves analogues of the Schur orthogonality relations that are analogous to formula (2.10). Ikeda [Ike1953], and Higman [Hig1955], following work of Gaschütz [Gas1952], construct "Casimir" type elements similar to those in (2.9) and §3 Ex. 7. In [CRe1981] §9 Curtis and Reiner use a similar approach but with different proofs, communicated to them by R. Kilmoyer, to obtain theorems (3.8) and (3.9) for split semisimple algebras (over fields of characteristic 0). N. Wallach has told me that essentially the same approach works for finite dimensional Lie algebras.
This approach is useful for studying semisimple algebras that have distiguished bases. The recent interest in quantum deformations is producing a host of examples of semisimple algebras that are not group algebras but that do have distinguished bases. Some examples are Hecke algebras associated to root systems, the Brauer algebra, and the Birman-Wenzl algebra [BWe1989]. For an approach to the Hecke algebras that is essentially an application of the general theory given here see [GUn1989] and [Cr3] §68C.
I would like to thank Prof. A. Garsia for suggesting that I try to find an analogue of the symmetrization process for finite groups for the Brauer algebra. It was this problem that resulted in my discovery of this approach. I would like to thank Prof. C.W. Curtis for his helpful suggestions in locating literature with a similar approach. I would also like to thank Prof. Garsia for showing me the proofs of Exs. 4 and 5.
This is an excerpt from the unpublished first chapter of Arun Ram's dissertation entitled Representation Theory, written July 4, 1990.