Double centralizer nonsense

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

Last update: 6 November 2012

Double centralizer nonsense

Tensor products

If P and Q are two matrices with entries from , then the tensor product of P and Q is the matrix

PQ= pijQ, (4.1)

where pij denotes the (i,j)th entry in P. If V and W are two vector spaces with bases BV={vi} and BW={wi} respectively, the tensor product VW is the vector space consisting of the linear span of the words viwj. If V is dimension n and W is dimension nm. then VW is dimension nm. In general, for any vV and wW, the word vw can be expressed in terms of the words viwj by using linearity, i.e. for all c,d vi,vjBV and wr,wsBW,

(cvi+dvj) wr = cviwr+ dvjwr and vi (cwr+dws) = cviwr+ dviws.

Suppose that A and C are two arbitrary algebras. We can define an algebra structure on the vector space AC (we distinguish the tensor product of algebras from the vector space case by writing (a,c) instead of ac for a word in AC, aA, cC) by defining multiplication of elements of AC by

(a1,c1) (a2,c2) = (a1a2,c1c2) , (4.2)

for all a1,a2A and c1,c2C, amd extending linearly.

Suppose that V and W are representations of A and C respectively. Define an action of AC on the vector space VW by

(a,c)(vw) =(av)(cw) (4.3)

for all (a,c) and vw, aA, cC, vV, wW. This defines a representation of AC on VW under which the action of (a,c), aA, cC on VW is given by the matrix


Centralizer of a completely decomposable representation

Let V be a completely decomposable representation of an algebra A. Assume that

Vλ=1n Wλmλ,

where the Wi are nonisomorphic irreducible representations of V. This means that we can decompose V into irreducible subspaces Vλjm 1λn, 1jmλ, so that

V=λ,j Vλj,

where for each λ and j, VλjWλ. Let dλ=dimWi. Choosing a basis on each of the Vλj gives a basis of V which we denote . Using the basis of V, the algebra of the representation V is

V(A)= ( Im1 (Md1()) 0 0 0 0 Im2 (Md2()) 0 0 0 0 0 Imn (Mdm()) ) . (4.4)

(1.11) shows that the algebra of matrices that commute with all matrices in V(A), is

V(A) = ( Mm1 (W1(A)) 0 0 0 0 Mm2 (W2(A)) 0 0 0 0 0 Mmn (Wn(A)) ) .

Since, by Schur's Lemma, Wλ(A)= Idλ(), such that

V(A) = ( Mm1 (Id1()) 0 0 0 0 Mm2 (Id2()) 0 0 0 0 0 Mmn (Idn()) ) . (4.5)

(4.6) Theorem. If a representation V of an algebra A is completely decomposable in the form

Vλ=1n Wλmλ,

where the Wλ are nonisomorphic irreducibles, then the centralizer V(A) of V(A) is semsimple and

V(A) λ=1n Mmλ().


By a change of basis on V we can put the matrices of (4.5) in the form

( Id1 (Mm1()) 0 0 0 0 Id2 (Mm2()) 0 0 0 0 0 Idn (Mmn()) ) . (4.7)

These matrices are of exactly the same form as those in (4.4) except that the dλs and mλs are switched!! (4.7) shows that V(A) λ=1n Mmλ() as algebras.

Let B be an algebra with an action on B such that V(B)= V(A). Let B be the kernel of the act B on V and let C be the quotient B/B so that the induced action of C on V is injective. CV(B)= V(A).

From (4.5) we see that with respect to the basis on V the action of an element qC is given by a matrix of the form

( Q1 Im1 0 0 0 0 Q2 Im2 0 0 0 0 0 Qn Imn ) . (4.8)

where QλMmλ(). This action determines a map

ϕ: C λMmλ() q ( Q1 0 0 0 Q2 0 0 0 Qn ) ,

which, by Theorem (4.6), is an isomorphism. Note that, for each λ, the map

Cλ: C Mmλ() q Qλ (4.9)

is an irreducible representation of C.

Let Eijλ denote the matrix in λMmλ() that is 1 in the (i,j)th entry of the λth block and 0 everywhere else. Define a set of matrix units eijλ, 1λn, 1i,jmλ in C by

eijλ= ϕ-1 (Eijλ).

The action of the element eiiλ on V, is given by the matrix Eiiλ Im V(A). The action of this matrix on V is the projection p:VVλi;

Vλi= eiiλV.

Conversely, if {eijλ} is a set of matrix units of C, then, since 1=λ,ieiiλ as an element of C, we have a decomposition

V=1·V= ( λ,i eiiλ ) V=λ,i eiiλV.

Since the action of A on V commutes with the action of C we have that aeiiλV= eiiλaV eiiλV for all aA, showing that each of the spaces eiiλV is A-invariant. Since, §1 Ex. 5, eiiλVejjμ= unless λ=μ and i=j, the decomposition given above is a direct sum decomposition of V. This decomposition is a decomposition of V into irreducible subspaces under the action of A,

V=λ,i eiiλV. (4.10)

Define an action of CA on V by


where (q,a)CA and vV. Since the actions of C and A on V commute this action is well defined and makes V into an CA representation. Theorem (4.6) shows that the irreducible representations of C are in one to one correspondence with the irreducible representations of A appearing in the decomposition of V. Let Cλ denote the irreducible representation of C corresponding to A.

(4.11) Theorem. As CA representations,

Vλ=1n CλWλ.


With respect to the basis of V the action of (q,a)CA on V is given by the matrix product

( Q1 Im1 0 0 0 0 Q2 Im2 0 0 0 0 0 Qn Imn ) ( Im1 W1(a) 0 0 0 0 Im2 W2(a) 0 0 0 0 0 Imn Wn(a) ) ,

which is equal to

( Q1 W1(a) 0 0 0 0 Q2 W2(a) 0 0 0 0 0 Qn Wn(a) ) . (4.12)

Recalling (4.9) we see that the action of each block of (4.12) is by the representation CλWλ.


  1. Let G be a group and let V and W be two representations of G . Define an action of G on the vector space VW by g vw = gv gw , for all gG,vV and wW (see also Section 5 Ex 4). In matrix form, the representation VW is given by setting V d W g =V g W g , for each gG. Note, however, that if we extend this action to an action of A=G on VW, then for a general aA, a vw is not equal to av aw and V d W a is not equal to V a W a .
  2. Theorem 4.6 gives that there is a one-to-one correspondence between minimal central idempotents z λ C of C and characters χ A λ of irreducible representations of A of A appearing in the decomposition of V . Let χ C λ be the irreducible characters of C and for each λ set d λ C = χ C λ 1 , so that the d λ are the dimensions of the irreducible representations of C. The Frobenius map is the map F: Z C R A 1 d λ C z λ X χ A λ . Let t:CA be the trace of the action of CA on the representation V. By taking traces on each side of the isomorphism in Theorem 4.11 we have that t qa = λ χ C λ q χ A λ a . Let t C = t λ C be a nondegenerate trace on C , let B be a basis of C and for each gB let g* be the element of the dual bsis to B with respect to the trace t C such that t C gg* =1. Then, for any zZ C , the center of C, F z = gB t C zg* t g. , since, using 3.8 and 3.9, F z μ C d μ C = g 1 d μ C t C z μ C g* t g. = g tμC d μ C χ C μ g* t g. = g tμC d μ C χ C μ g* λ χ C λ q χ A λ . = g tμC d μ C δ μλ dλC t λ C χ A λ . = χ μ A . .

    If we apply the inverse F -1 of the Frobenius map to (4.13) we get F -1 t q. = λ χ C λ q zλC d λ C . Formula 3.13 shows that F -1 t q. = λ t λ C dλ C z λ C q . In the case that t C is the trace of the regular representation λ t λ C dλ C z λ C =1 and F -1 t q. = q .

Notes and References

"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 Vf, dimV=n of Gl(n), Schur-Weyl duality. This was the key concept in Schur's original work on the rational representations of Gl(n).

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.

Notes and References

This is an excerpt from the unpublished first chapter of Arun Ram's dissertation entitled Representation Theory, written July 4, 1990.

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