The Unapologetic Mathematician

Mathematics for the interested outsider

Dimensions of Hom Spaces

Now that we know that hom spaces are additive, we’re all set to make a high-level approach to generalizing last week’s efforts. We’re not just going to deal with endomorphism algebras, but with all the \hom-spaces.

Given G-modules V and W, Maschke’s theorem tells us that we can decompose our representations as

\displaystyle\begin{aligned}V&\cong m_1V^{(1)}\oplus\dots\oplus m_kV^{(k)}\\W&\cong n_1V^{(1)}\oplus\dots\oplus n_kV^{(k)}\end{aligned}

where the V^{(i)} are pairwise-inequivalent irreducible G-modules with degrees d_i. I’m including all the irreps that show up in either decomposition, so some of the coefficients m_i or n_i may well be zero. This is not a problem, since it just means direct-summing on a trivial module.

So let’s use additivity! We find

\displaystyle\begin{aligned}\hom_G(V,W)&\cong\hom_G\left(\bigoplus\limits_{i=1}^km_iV^{(i)},\bigoplus\limits_{j=1}^kn_jV^{(j)}\right)\\&=\bigoplus\limits_{i=1}^k\bigoplus\limits_{j=1}^k\hom(m_iV^{(i)},n_jV^{(j)})\end{aligned}

Now to calculate these summands, we can pick a basis for V^{(i)} and V^{(j)} and use the same sorts of methods we did to calculate commutant algebras. We find that if i\neq j — V^{(i)}\not\cong V^{(j)} — then there are no G-morphisms at all, even if we include multiplicities. On the other hand, if i=j we find that an intertwinor between m_iV^{(i)} and n_iV^{(i)} has the form M_{m_i,n_i}\boxtimes I_{d_i}, where M_{m_i,n_i} is an m_i\times n_i complex matrix. That is, as a vector space it’s isomorphic to the space of m_i\times n_i matrices.

We conclude

\displaystyle\hom_G(V,W)\cong\bigoplus\limits_{i=1}^k\mathrm{Mat}_{m_i,n_i}(\mathbb{C})

and its dimension is

\displaystyle\dim\hom_G(V,W)=\sum\limits_{i=1}^k\dim\mathrm{Mat}_{m_i,n_i}(\mathbb{C})=\sum\limits_{i=1}^km_in_i

Notice that any i for which m_i=0 or n_i=0 doesn’t count for anything.

As a special case, we consider the endomorphism algebra \mathrm{End}_G(V)=\hom_G(V,V). This time we assume that none of the m_i are zero. We find:

\displaystyle\mathrm{End}_G(V)\cong\bigoplus\limits_{i=1}^k\mathrm{Mat}_{m_i}(\mathbb{C})

with dimension

\displaystyle\dim\mathrm{End}_G(V)=\sum\limits_{i=1}^k\dim\mathrm{Mat}_{m_i}(\mathbb{C})=\sum\limits_{i=1}^km_i^2

Just like before, we can calculate the center, which goes summand-by-summand. Each summand is (isomorphic to) a complete matrix algebra, so we know that its center is isomorphic to \mathbb{C}. Thus we find that the center of \mathrm{End}_G(V) is the direct sum of k copies of \mathbb{C}, and so has dimension k.

As one last corollary, let V=V^{(1)} be irreducible and let W be any representation. Then we calculate the dimension of the \hom-space:

\displaystyle\begin{aligned}\dim\hom_G(V,W)&=\dim\bigoplus\limits_{i=1}^k\hom_G(V^{(1)},n_iV^{(i)})\\&=\sum\limits_{i=1}^k\dim\hom_G(V^{(1)},n_iV^{(i)})\\&=n_1\end{aligned}

That is, the dimension of the space of intertwinors is exactly the multiplicity of V^{(1)} in the representation W.

October 12, 2010 Posted by | Algebra, Group theory, Representation Theory | 5 Comments

   

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