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 where is the standard representation of .

Such a tensor product has no invariants unless 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 . 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 representations, , 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 are also Howe dual — this is obvious and does not tell us anything. But also their actions on are Howe dual for any k. Moreover, we have the following expression for the decomposition where ranges over all partitions with k boxes and at most n rows and m columns and 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 where 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 as and we can decompose . Here the direct sum ranges over all which sum to . Now, the key is that this decomposition is as representations! In fact it is exactly the decomposition into weight spaces for T!

Hence if we are interested in , then we should just look at the weight space in where is a rectangle with p rows and n columns. This is exactly what we wanted.

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