Last update: 6 November 2012
If and are two matrices with entries from then the tensor product of and is the matrix
where denotes the entry in If and are two vector spaces with bases and respectively, the tensor product is the vector space consisting of the linear span of the words If is dimension and is dimension then is dimension In general, for any and the word can be expressed in terms of the words by using linearity, i.e. for all and
Suppose that and are two arbitrary algebras. We can define an algebra structure on the vector space (we distinguish the tensor product of algebras from the vector space case by writing instead of for a word in by defining multiplication of elements of by
for all and amd extending linearly.
Suppose that and are representations of and respectively. Define an action of on the vector space by
for all and This defines a representation of on under which the action of on is given by the matrix
Let be a completely decomposable representation of an algebra Assume that
where the are nonisomorphic irreducible representations of This means that we can decompose into irreducible subspaces so that
where for each and Let Choosing a basis on each of the gives a basis of which we denote Using the basis of the algebra of the representation is
(1.11) shows that the algebra of matrices that commute with all matrices in is
Since, by Schur's Lemma, such that
(4.6) Theorem. If a representation of an algebra is completely decomposable in the form
where the are nonisomorphic irreducibles, then the centralizer of is semsimple and
By a change of basis on we can put the matrices of (4.5) in the form
These matrices are of exactly the same form as those in (4.4) except that the and are switched!! (4.7) shows that as algebras.
Let be an algebra with an action on such that Let be the kernel of the act on and let be the quotient so that the induced action of on is injective.
From (4.5) we see that with respect to the basis on the action of an element is given by a matrix of the form
where This action determines a map
which, by Theorem (4.6), is an isomorphism. Note that, for each the map
is an irreducible representation of
Let denote the matrix in that is 1 in the entry of the block and 0 everywhere else. Define a set of matrix units in by
The action of the element on is given by the matrix The action of this matrix on is the projection
Conversely, if is a set of matrix units of then, since as an element of we have a decomposition
Since the action of on commutes with the action of we have that for all showing that each of the spaces is Since, §1 Ex. 5, unless and the decomposition given above is a direct sum decomposition of This decomposition is a decomposition of into irreducible subspaces under the action of
Define an action of on by
where and Since the actions of and on commute this action is well defined and makes into an representation. Theorem (4.6) shows that the irreducible representations of are in one to one correspondence with the irreducible representations of appearing in the decomposition of Let denote the irreducible representation of corresponding to
(4.11) Theorem. As representations,
With respect to the basis of the action of on is given by the matrix product
which is equal to
Recalling (4.9) we see that the action of each block of (4.12) is by the representation
If we apply the inverse of the Frobenius map to (4.13) we get Formula 3.13 shows that In the case that is the trace of the regular representation and
"Double centralizer nonsense" is a term that has been used by R. Stanley in reference to Theorems (4.6) and (4.12). I have chosen to adopt this term as well. These results are originally due to I. Schur [Scl],[Sch1927], and are often referred to as the Double Commutant Theorem, or, in the special case of the representation of Schur-Weyl duality. This was the key concept in Schur's original work on the rational representations of
The Frobenius map given in Ex. 3 is a generalization of the classical Frobenius map [Mac1979] §1.7. In a paper [Fro1900] that demonstrates absolute genius, Frobenius used it as a tool for determining the characters of the symmetric groups.
This is an excerpt from the unpublished first chapter of Arun Ram's dissertation entitled Representation Theory, written July 4, 1990.