The ubiquity of Howe duality

I’m been thinking recently about multiple tensor products of SL_n representations and I keep running into the strange phenomenon of Howe duality. Consider in particular the problem of finding the dimension of the space of SL_n invariants inside a tensor product \Lambda^{k_1} V \otimes \cdots \otimes \Lambda^{k_m} V where V = \mathbb{C}^n is the standard representation of SL_n.

Such a tensor product has no invariants unless k_1 + 2k_2 + \ldots + mk_m =  pn for some integer p and in this case, the dimension is the number of semistandard young tableaux of shape an n x p rectangle filled with k_1 1s, k_2 2s, ..., k_m ms . This can be proven using the theory of Littelmann paths, but the reader will probably as recognize that this is also the dimension of a certain weight space of a certain representation of GL_m. This strange coincidence can be explained using Howe duality.

In general, let V be a vector space with commuting actions of two (finite or reductive) groups G and H. We say these actions are Howe dual if the groups are each others commutant. In this case, it is easy to prove that there is a decomposition as G\times H representations, V = \oplus A_i \otimes B_i , where A_i, B_i are irreducible representations of G and H respectively.

Now, consider two vector spaces V, W (of dimensions n and m respectively) and associated general linear groups GL(V) and GL(W). The actions of GL(V) and GL(W) on V \otimes W are also Howe dual — this is obvious and does not tell us anything. But also their actions on \Lambda^k(V \otimes W) are Howe dual for any k. Moreover, we have the following expression for the decomposition \Lambda^k(V \otimes W) = \oplus_{\lambda} V_\lambda \otimes W_{\lambda^\vee} where \lambda ranges over all partitions with k boxes and at most n rows and m columns and V_\lambda denotes the appropriate Schur functor applied to V.

If we restrict to SL(V) = SL_n, and assume that k = np for some p, then this tells us that Hom_{SL(V)} (triv, \Lambda^k(V \otimes W) ) = W_{\lambda} where \lambda is a rectangle with p rows and n columns. This is almost what we want.

To go further, we should fix a direct sum decomposition of W into one dimensional subspaces, or equivalently a maximal torus T in GL(W). Then we regard V \otimes W as V \oplus \cdots \oplus V and we can decompose \Lambda^k(V \otimes W) = \oplus \Lambda^{k_1} V \otimes \cdots \otimes \Lambda^{k_m} V. Here the direct sum ranges over all (k_1, \dots, k_m) which sum to k . Now, the key is that this decomposition is as GL(V) \times T representations! In fact it is exactly the decomposition into weight spaces for T!

Hence if we are interested in Hom_{SL(V)} (triv,  \Lambda^{k_1} V \otimes \cdots \otimes \Lambda^{k_m} V)  , then we should just look at the (k_1, \dots, k_m) weight space in W_{\lambda} where \lambda is a rectangle with p rows and n columns. This is exactly what we wanted.