## 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 -spaces.

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

where the are pairwise-inequivalent irreducible -modules with degrees . I’m including all the irreps that show up in either decomposition, so some of the coefficients or 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

Now to calculate these summands, we can pick a basis for and and use the same sorts of methods we did to calculate commutant algebras. We find that if — — then there are no -morphisms at all, even if we include multiplicities. On the other hand, if we find that an intertwinor between and has the form , where is an complex matrix. That is, as a vector space it’s isomorphic to the space of matrices.

We conclude

and its dimension is

Notice that any for which or doesn’t count for anything.

As a special case, we consider the endomorphism algebra . This time we assume that none of the are zero. We find:

with dimension

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 . Thus we find that the center of is the direct sum of copies of , and so has dimension .

As one last corollary, let be irreducible and let be any representation. Then we calculate the dimension of the -space:

That is, the dimension of the space of intertwinors is exactly the multiplicity of in the representation .

[…] something interesting to notice in our formulæ for the dimensions of spaces of intertwinors: they’re symmetric between the two […]

Pingback by Hom Space Duals « The Unapologetic Mathematician | October 13, 2010 |

[…] that this is very similar to the result we showed at the end of calculating the dimensions of spaces of morphisms. This is not a […]

Pingback by Consequences of Orthogonality « The Unapologetic Mathematician | October 25, 2010 |

[…] irreps stay distinct, equivalent ones lose their separate identities in the sum. Indeed, we’ve seen […]

Pingback by Subspaces from Irreducible Representations « The Unapologetic Mathematician | November 10, 2010 |

[…] we know that , and thus the dimension of our space of invariants is the dimension of the space. We’ve seen that this is the multiplicity of the trivial representation in , which we’ve also seen is the […]

Pingback by The Dimension of the Space of Tensors Over the Group Algebra « The Unapologetic Mathematician | November 16, 2010 |

[…] the Specht modules are irreducible, we know that the dimension of this space is the multiplicity of in . And the dimension, of course, is the […]

Pingback by Kostka Numbers « The Unapologetic Mathematician | February 17, 2011 |