## Lectures in Representation Theory

Last update: 27 August 2013

## Lecture 20

More generally, let $\stackrel{ˆ}{V}$ be a finite index set and suppose ${A}_{\lambda }\subseteq {M}_{{d}_{\lambda }}\left(ℂ\right),$ where ${d}_{\lambda }\in ℙ$ for each $\lambda \in \stackrel{ˆ}{V}\text{.}$ Let $d={\sum }_{\lambda }{d}_{\lambda }$ and let $A={⨁}_{\lambda \in \stackrel{ˆ}{V}}{A}_{\lambda }\text{.}$

Elements of $A$ are matrices of the form $( aλ aμ 0 0 ⋱ aδ )$ where ${a}_{\rho }\in {A}_{\rho }$ is a ${d}_{\rho }×{d}_{\rho }$ matrix with complex entries. So $A$ is a subalgebra of ${M}_{d}\left(ℂ\right)$ consisting of block diagonal matrices. Note that we do not insist that ${A}_{\lambda }={M}_{{d}_{\lambda }}\left(ℂ\right),$ so $A$ is merely block diagonal and not necessarily semisimple.

Proposition 3.13 The centralizer of an algebra $A$ of block diagonal matrices is the direct sum of the centralizers of each of the blocks. That is, $⨁λ∈VˆAλ‾= ⨁λ∈Vˆ Aλ‾.$

 Proof. Since block diagonal matrices multiply componentwise, it is obvious that $\stackrel{‾}{\left({\oplus }_{\lambda \in \stackrel{ˆ}{V}}{A}_{\lambda }\right)}\supseteq {⨁}_{\lambda \in \stackrel{ˆ}{V}}\stackrel{‾}{{A}_{\lambda }}\text{.}$ Suppose, on the other hand, that $B={\sum }_{\mu ,\nu \in \stackrel{ˆ}{V}}{E}_{\mu ,\nu }\otimes {b}_{\mu ,\nu }\in \stackrel{‾}{A}\text{.}$ Elements of $A$ may be written as ${\sum }_{\lambda \in \stackrel{ˆ}{V}}{E}_{\lambda ,\lambda }\otimes {a}_{\lambda }$ where ${a}_{\lambda }\in {A}_{\lambda }\text{.}$ Then $B$ commutes with all such elements, hence $(∑λ∈VˆEλ,λ⊗aλ) (∑μ,ν∈VˆEμ,ν⊗bμ,ν) = (∑μ,ν∈VˆEμ,ν⊗bμ,ν) (∑λ∈VˆEλ,λ⊗aλ) .$ Multiplying out, we obtain $∑λ,ν∈Vˆ Eλ,ν⊗aλ bλ,ν= ∑μ,λ∈Vˆ Eμ,λ⊗ bμ,λaλ.$ Comparing coefficients of ${E}_{\lambda ,\nu }$ on each side, we obtain $aλvλ,ν= bλ,νaν (3.14)$ for all $\lambda ,\nu \in \stackrel{ˆ}{V}$ and for all choices of elements ${a}_{\mu }\in {A}_{\mu }\text{.}$Taking $\mu =\lambda$ we obtain $aλbλ,λ= bλ,λaλ$ for all ${a}_{\lambda }\in {A}_{\lambda }\text{;}$ hence, ${b}_{\lambda ,\lambda }\in \stackrel{‾}{{A}_{\lambda }}$ for all $\lambda \in \stackrel{ˆ}{V}\text{.}$ Fix $\lambda \in \stackrel{ˆ}{V}$ and choose ${a}_{\lambda }=I$ and ${a}_{\mu }=0$ for $\mu \ne \lambda \text{.}$ Then 3.14 becomes $bλ,ν= aλbλ,ν =bλ,νaν =0$ so ${b}_{\lambda ,\nu }=0$ whenever $\lambda \ne \nu \text{.}$ Thus $B={\sum }_{\mu \in \stackrel{ˆ}{V}}{E}_{\mu ,\mu }\otimes {b}_{\mu }$ where ${b}_{\mu }\in \stackrel{‾}{{A}_{\mu }}\text{.}$ Therefore, $\stackrel{‾}{A}\subseteq {\oplus }_{\lambda \in \stackrel{ˆ}{V}}\stackrel{‾}{{A}_{\lambda }}\text{.}$ $\square$

Corollary 3.15 If $\stackrel{ˆ}{A}$ is a finite index set, then $⨁λ∈AˆMdλ(ℂ)‾= ⨁λ∈Aˆ Mdλ(ℂ)‾ =⨁λ∈Aˆℂ Idλ$

Let $V$ be a representation of an arbitrary (associative) algebra $A$ and suppose $V$ is completely decomposable. Denote the decomposition of $V$ by $V≅⨁λ∈Vˆ (Vλ)⊗mλ =∑λ∈Vˆmλ Vλ$ where the ${V}^{\lambda }$ denote, as usual, representatives of distinct isomorphism classes of irreducible $A\text{-modules.}$ Let ${d}_{\lambda }=\text{dim} {V}^{\lambda }\text{.}$

If we choose a basis ${ℬ}_{\lambda }=\left\{{w}_{1}^{\lambda },{w}_{2}^{\lambda },\dots {w}_{{d}_{\lambda }}^{\lambda }\right\}$ for each irreducible ${V}^{\lambda },$ then we may choose a basis for $V$ by taking unions of ${m}_{\lambda }$ copies of each ${ℬ}_{\lambda }\text{.}$ With respect to this basis, the matrices of $V\left(A\right)$ are of the form $V(a)=⨁λ∈Vˆ ( Vλ(a) Vλ(a) 0 0 ⋱ Vλ(a) )$ where there are ${m}_{\lambda }$ many blocks of ${V}^{\lambda }$ for each $\lambda \text{.}$ Thus, in each matrix $V\left(a\right),$ each $\lambda$ block consists of the same matrix repeated ${m}_{\lambda }$ times, and so $V(A)≅⊕λ Imλ(Vλ(A)) (3.16)$ as algebras.

From the previous propositions, $V(A)‾ = ⨁λ∈Vˆ Imλ (Vλ(A)) ‾ = ⨁λ∈Vˆ Imλ (Vλ(A)) ‾ = ⨁λ∈Vˆ Mmλ (Vλ(A)‾) = ⨁λ∈Vˆ Mmλ (Idλ(ℂ))$ So the $\lambda$ block of a matrix $b$ in the centralizer $\stackrel{‾}{V\left(A\right)}$ consists of ${m}_{\lambda }^{2}$ scalar matrices ${c}_{i,j}^{\lambda }{I}_{{d}_{\lambda }}\text{.}$ That is, for $b\in \stackrel{‾}{V\left(a\right)},$ $b=⨁λ∈Vˆ ( c1,1λ c1,1λ ⋱ c1,1λ ⋯ c1,mλλ c1,mλλ ⋱ c1,mλλ ⋮ ⋱ ⋮ cmλ,1λ cmλ,1λ ⋱ cmλ,1λ ⋯ cmλ,mλλ cmλ,mλλ ⋱ cmλ,mλλ )$

We next perform a (somewhat complicated) reordering of the basis of the module $V\text{.}$ In terms of representations, this exchanges $V$ for the equivalent representation $a↦{P}^{-1}V\left(a\right)P$ where $P$ is a product of elementary matrices describing the permutation of the basis.

Recall that interchanging two columns $i$ and $j$ of a matrix is equivalent to multiplying on the right by the elementary matrix $E={\sum }_{k\ne i,j}{E}_{k,k}+{E}_{i,j}+{E}_{j,i}\text{.}$ Multiplying by this elementary matrix on the right interchanges rows $i$ and $j\text{.}$ Note that this matrix is self-invertible, $E={E}^{-1},$ and so conjugating a matrix $X$ by $E$ (which is equivalent to interchanging the $i\text{th}$ and $j\text{th}$ basis vectors) corresponds to interchanging the $i\text{th}$ and $j\text{th}$ rows and columns of $X\text{.}$ We may thus describe the permutation of the basis and the resulting change on $V\left(A\right)$ and $\stackrel{‾}{V\left(A\right)}$ is to similarly permute their rows and columns.

We number the columns of blocks, starting from the left, from 1 to ${m}_{\lambda }\text{.}$ There are ${d}_{\lambda }$ columns within a particular column of blocks, which we label 1 up to ${d}_{\lambda }$ from left to right. Hence, the $n\text{th}$ column of $m\text{th}$ block of the $\lambda$ component may be labeled by the ordered pair $\left(m,n\right)\text{.}$ Since the individual blocks are square, we may label the rows similarly using the number of their block and row within that block. Thus $m$ ranges from 1 to ${m}_{\lambda }$ and $n$ ranges from 1 to ${d}_{\lambda }$ and the $\left\{\left({m}_{1},{n}_{1}\right),\left({m}_{2},{n}_{2}\right)\right\}$ entry of the $\lambda$ component is ${\delta }_{{n}_{1},{n}_{2}}{c}_{{m}_{1},{m}_{2}}^{\lambda }\text{.}$

Permute the columns so that they appear in the order $\left(1,1\right),$ $\left(2,1\right),$ $\left(3,1\right),$ $\dots ,$ $\left({m}_{\lambda },1\right),$ $\left(1,2\right),$ $\left(2,2\right),$ $\dots ,$ $\left({m}_{\lambda },{d}_{\lambda }\right)$ (i.e. interchange the block number and column number). After permuting the columns, but before performing the similar permutation of the rows, the $\lambda$ component of $b$ is of the form $( c1,1 c1,2 … c1,mλ c1,1 c1,2 … c1,mλ … c1,1 c1,2 … c1,mλ c2,1 c2,2 … c2,mλ c2,1 c2,2 … c2,mλ … c2,1 c2,2 … c2,mλ ⋱ cmλ,1 cmλ,2 … cmλ,mλ cmλ,1 cmλ,2 … cmλ,mλ … cmλ,1 cmλ,2 … cmλ,mλ )$

It should be noted that after rearranging columns, the obvious block structure no longer consists of square blocks. Each block above is a ${d}_{\lambda }$ by ${m}_{\lambda }$ matrix. Therefore, the rows of the $\lambda$ component are still labeled by $\left(m,n\right)$ with $1\le m\le {m}_{\lambda }$ and $1\le n\le {d}_{\lambda }\text{.}$

If we now perform the permutation of the rows so that they are ordered $\left(1,1\right),$ $\left(2,1\right),$ $\left(3,1\right),$ $\dots ,$ $\left({d}_{\lambda },{m}_{\lambda }\right),$ observe that we will collect all of the rows with nonzero entries in the first column of blocks into the first ${m}_{\lambda }$ rows. In fact, the $\lambda$ component becomes $( Cλ Cλ ⋱ Cλ ) =Idλ⊗Cλ$ where ${C}^{\lambda }=\left({c}_{i,j}^{\lambda }\right)$ is an ${m}_{\lambda }×{m}_{\lambda }$ matrix.

Since $b$ is block diagonal, we may perform this reordering simultaneously on each component. Hence, after this reordering of the basis, we may regard $b={\oplus }_{\lambda }{I}_{{d}_{\lambda }}\otimes {C}^{\lambda }\text{.}$ We have shown that $\stackrel{‾}{V\left(a\right)}\cong {\oplus }_{\lambda }{I}_{{d}_{\lambda }}\left({M}_{{m}_{\lambda }}\left(ℂ\right)\right),$ but for any algebra $A,$ ${I}_{{d}_{\lambda }}\left(A\right)\cong A\text{.}$

Proposition 3.17 Let $A$ be an arbitrary associative algebra and let $V$ be a completely decomposable finite dimensional representation of $A\text{.}$ If $V={\oplus }_{\lambda }{m}_{\lambda }{V}^{\lambda }$ is a decomposition of $V$ into pairwise nonisomorphic irreducible modules, then $V(A)‾≅ ⨁λ∈Vˆ Mmλ(ℂ)$ where ${d}_{\lambda }=\text{dim} {V}^{\lambda }\text{.}$

In particular, $\stackrel{‾}{V\left(A\right)}$ is semisimple and $\text{dim} \stackrel{‾}{V\left(A\right)}={\sum }_{\lambda }{m}_{\lambda }^{2},$ where ${m}_{\lambda }$ is the multiplicity of ${V}^{\lambda }$ in $V\text{.}$

We have fully described $\stackrel{‾}{V\left(A\right)}$ in terms of complex matrices, which already yields the useful information that centralizer algebras of arbitrary finite dimensional representations are semisimple. In 3.16, we showed that $V\left(A\right)={\oplus }_{\lambda \in \stackrel{ˆ}{V}}{I}_{{m}_{\lambda }}\left({V}^{\lambda }\left(A\right)\right)$ and would like to simplify the ${V}^{\lambda }\left(A\right)$ to matrices.

## 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.