## Double centralizer nonsense

Last update: 6 November 2012

## Double centralizer nonsense

### Tensor products

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

$P⊗Q= ∣pijQ∣, (4.1)$

where ${p}_{ij}$ denotes the $\left(i,j\right)\text{th}$ entry in $P\text{.}$ If $V$ and $W$ are two vector spaces with bases ${B}_{V}=\left\{{v}_{i}\right\}$ and ${B}_{W}=\left\{{w}_{i}\right\}$ respectively, the tensor product $V\otimes W$ is the vector space consisting of the linear span of the words ${v}_{i}{w}_{j}\text{.}$ If $V$ is dimension $n$ and $W$ is dimension $nm\text{.}$ then $V\otimes W$ is dimension $nm\text{.}$ In general, for any $v\in V$ and $w\in W,$ the word $vw$ can be expressed in terms of the words ${v}_{i}{w}_{j}$ by using linearity, i.e. for all $c,d\in ℂ$ ${v}_{i},{v}_{j}\in {B}_{V}$ and ${w}_{r},{w}_{s}\in {B}_{W},$

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

Suppose that $A$ and $C$ are two arbitrary algebras. We can define an algebra structure on the vector space $A\otimes C$ (we distinguish the tensor product of algebras from the vector space case by writing $\left(a,c\right)$ instead of $ac$ for a word in $A\otimes C,$ $a\in A,$ $c\in C\text{)}$ by defining multiplication of elements of $A\otimes C$ by

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

for all ${a}_{1},{a}_{2}\in A$ and ${c}_{1},{c}_{2}\in C,$ amd extending linearly.

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

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

for all $\left(a,c\right)$ and $vw,$ $a\in A,$ $c\in C,$ $v\in V,$ $w\in W\text{.}$ This defines a representation of $A\otimes C$ on $V\otimes W$ under which the action of $\left(a,c\right),$ $a\in A,$ $c\in C$ on $V\otimes W$ is given by the matrix

$V(a)⊗W(c).$

### Centralizer of a completely decomposable representation

Let $V$ be a completely decomposable representation of an algebra $A\text{.}$ Assume that

$V≅⊕λ=1n Wλ⊕mλ,$

where the ${W}_{i}$ are nonisomorphic irreducible representations of $V\text{.}$ This means that we can decompose $V$ into irreducible subspaces ${V}_{\lambda j}m$ $1\le \lambda \le n,$ $1\le j\le {m}_{\lambda },$ so that

$V=⊕λ,j Vλj,$

where for each $\lambda$ and $j,$ ${V}_{\lambda j}\cong {W}_{\lambda }\text{.}$ Let ${d}_{\lambda }=\text{dim}\phantom{\rule{0.2em}{0ex}}{W}_{i}\text{.}$ Choosing a basis on each of the ${V}_{\lambda j}$ gives a basis of $V$ which we denote $ℬ\text{.}$ Using the basis $ℬ$ of $V,$ the algebra of the representation $V$ is

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

(1.11) shows that the algebra of matrices that commute with all matrices in $V\left(A\right),$ 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, $\stackrel{‾}{{W}_{\lambda }\left(A\right)}={I}_{{d}_{\lambda }}\left(ℂ\right),$ such that

$V(A)‾ = ( Mm1 (Id1(ℂ)) 0 0 … 0 0 Mm2 (Id2(ℂ)) 0 … 0 … … … … … 0 0 … 0 Mmn (Idn(ℂ)) ) . (4.5)$

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

$V≅⊕λ=1n Wλ⊕mλ,$

where the ${W}_{\lambda }$ are nonisomorphic irreducibles, then the centralizer $\stackrel{‾}{V\left(A\right)}$ of $V\left(A\right)$ is semsimple and

$V(A)‾≅ ⊕λ=1n Mmλ(ℂ).$

 Proof. By a change of basis on $V$ we can put the matrices of (4.5) in the form $( Id1 (Mm1(ℂ)) 0 0 … 0 0 Id2 (Mm2(ℂ)) 0 … 0 … … … … … 0 0 … 0 Idn (Mmn(ℂ)) ) . (4.7)$ These matrices are of exactly the same form as those in (4.4) except that the ${d}_{\lambda }\text{s}$ and ${m}_{\lambda }\text{s}$ are switched!! (4.7) shows that $\stackrel{‾}{V\left(A\right)}\cong {\oplus }_{\lambda =1}^{n}{M}_{{m}_{\lambda }}\left(ℂ\right)$ as algebras. $\square$

Let $B$ be an algebra with an action on $B$ such that $V\left(B\right)=\stackrel{‾}{V\left(A\right)}\text{.}$ Let ${B}^{\prime }$ be the kernel of the act $B$ on $V$ and let $C$ be the quotient $B/{B}^{\prime }$ so that the induced action of $C$ on $V$ is injective. $C\cong V\left(B\right)=\stackrel{‾}{V\left(A\right)}\text{.}$

From (4.5) we see that with respect to the basis $ℬ$ on $V$ the action of an element $q\in C$ 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 ${Q}_{\lambda }\in {M}_{{m}_{\lambda }}\left(ℂ\right)\text{.}$ This action determines a map

$ϕ: C ⟶ ⊕λMmλ(ℂ) q ⟼ ( Q1 0 … 0 0 Q2 … 0 … … … … 0 0 … Qn ) ,$

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

$Cλ: C ⟶ Mmλ(ℂ) q ⟼ Qλ (4.9)$

is an irreducible representation of $C\text{.}$

Let ${E}_{ij}^{\lambda }$ denote the matrix in ${\oplus }_{\lambda }{M}_{{m}_{\lambda }}\left(ℂ\right)$ that is 1 in the $\left(i,j\right)\text{th}$ entry of the $\lambda \text{th}$ block and 0 everywhere else. Define a set of matrix units ${e}_{ij}^{\lambda },$ $1\le \lambda \le n,$ $1\le i,j\le {m}_{\lambda }$ in $C$ by

$eijλ= ϕ-1 (Eijλ).$

The action of the element ${e}_{ii}^{\lambda }$ on $V,$ is given by the matrix ${E}_{ii}^{\lambda }\otimes {I}_{\stackrel{\to }{m}}\in \stackrel{‾}{V\left(A\right)}\text{.}$ The action of this matrix on $V$ is the projection $p:\phantom{\rule{0.2em}{0ex}}V\to {V}_{\lambda i};$

$Vλi= eiiλV.$

Conversely, if $\left\{{e}_{ij}^{\lambda }\right\}$ is a set of matrix units of $C,$ then, since $1=\sum _{\lambda ,i}{e}_{ii}^{\lambda }$ 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 $a{e}_{ii}^{\lambda }V={e}_{ii}^{\lambda }aV\subset {e}_{ii}^{\lambda }V$ for all $a\in A,$ showing that each of the spaces ${e}_{ii}^{\lambda }V$ is $A\text{-invariant.}$ Since, §1 Ex. 5, ${e}_{ii}^{\lambda }V\cap {e}_{jj}^{\mu }=\varnothing$ unless $\lambda =\mu$ and $i=j,$ the decomposition given above is a direct sum decomposition of $V\text{.}$ 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 $C\otimes A$ on $V$ by

$(q,a)v=qav$

where $\left(q,a\right)\in C\otimes A$ and $v\in V\text{.}$ Since the actions of $C$ and $A$ on $V$ commute this action is well defined and makes $V$ into an $C\otimes A$ 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 $V\text{.}$ Let ${C}_{\lambda }$ denote the irreducible representation of $C$ corresponding to $A\text{.}$

(4.11) Theorem. As $C\otimes A$ representations,

$V≅⊕λ=1n Cλ⊗Wλ.$

 Proof. With respect to the basis $ℬ$ of $V$ the action of $\left(q,a\right)\in C\otimes A$ 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 ${C}_{\lambda }\otimes {W}_{\lambda }\text{.}$ $\square$

### Examples

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 $V\otimes W$ by $g vw = gv gw ,$ for all $g\in G,v\in V$ and $w\in W$ (see also Section 5 Ex 4). In matrix form, the representation $V\otimes W$ is given by setting $V ⊗ d W g =V g ⊗W g ,$ for each $g\in G.$ Note, however, that if we extend this action to an action of $A=ℂG$ on $V\otimes W,$ then for a general $a\in A,$ $a\left(vw\right)$ is not equal to $\left(av\right)\left(aw\right)$ and $\left(V{\otimes }_{d}W\right)\left(a\right)$ is not equal to $V\left(a\right)\otimes W\left(a\right).$
2. Theorem 4.6 gives that there is a one-to-one correspondence between minimal central idempotents ${z}_{\lambda }^{C}$ of $C$ and characters ${\chi }_{A}^{\lambda }$ of irreducible representations of $A$ of $A$ appearing in the decomposition of $V$. Let ${\chi }_{C}^{\lambda }$ be the irreducible characters of $C$ and for each $\lambda$ set ${d}_{\lambda }^{C}={\chi }_{C}^{\lambda }\left(1\right),$ so that the ${d}_{\lambda }$ 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 $t:C\otimes A\to ℂ$ be the trace of the action of $C\otimes A$ 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 ${\stackrel{\to }{t}}_{C}=\left({t}_{\lambda }^{C}\right)$ be a nondegenerate trace on $C$, let $B$ be a basis of $C$ and for each $g\in B$ let $g*$ be the element of the dual bsis to $B$ with respect to the trace ${\stackrel{\to }{t}}_{C}$ such that ${\stackrel{\to }{t}}_{C}\left(gg*\right)=1.$ Then, for any $z\in Z\left(C\right),$ the center of $C,$ $F z = ∑ g∈B t → C zg* t g. ,$ 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 ${\stackrel{\to }{t}}_{C}$ is the trace of the regular representation $\sum _{\lambda }\left(\frac{{t}_{\lambda }^{C}}{{d}_{\lambda }^{C}}\right){z}_{\lambda }^{C}=1$ and ${F}^{-1}\left(t\left(q,.\right)\right)=\left[q\right].$

### 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 ${V}^{\otimes f},$ $\text{dim}\phantom{\rule{0.2em}{0ex}}V=n$ of $\text{Gl}\left(n\right),$ Schur-Weyl duality. This was the key concept in Schur's original work on the rational representations of $\text{Gl}\left(n\right)\text{.}$

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.