An interesting formula in combinatorial representation theory

Some research projects give you more than others in terms of reward and enjoyment. I’ve been very lucky to have been part of an enthusiastic team in Stephen Glasby, Luke Morgan, and Alice Niemeyer on a problem concerning automorphisms of p-groups; but more of that in a later post. In our investigations, we needed to know some basic data on polynomial representations of the general linear group GL(d,p). Consider the natural action of GL(d,p) on the tensor power T^n V where V is the vector space \mathbb{F}_p^d. It is a well-known result of Schur (and I won’t elaborate on it here) that if p>n, then we can parameterise the irreducible GL(d,p)-modules by the partitions of n. (Well actually, the characteristic 0 analogue is due to Schur, but it was folklore for a long time until a paper of Benson and Doty). For example, let us take the tensor square (n=2). If p is odd, then it is a classical fact that T^2 V breaks up into two smaller GL(d,p)-modules, namely

T^2 V \cong S^2 V \oplus A^2 V

where S^2 V is the symmetric square of V and A^2 V is the alternating square of V. The partitions here are the trivial partitions of the number 2. The partition (1,1) corresponds to A^2V and the partition (2,0) corresponds to S^2 V.

We are interested in something slightly more difficult. We actually want to know the irreducible constituents of the free Lie algebra L(V) generated by V. The connection between the two settings is cute. Define a bracket operation on the tensor algebra T(V) by [u,v] = u\otimes v-v\otimes u. Then we obtain a graded Lie algebra L(V)=\bigoplus_{n=1}^\infty L_n where each L_n is just L(V)\cap T^n(V). Each L_n then breaks up into irreducibles indexed by partitions of n, except we don’t know the multiplicities of each submodule. Write V^\lambda for the GL(V)-module corresponding to a partition \lambda of n. Note that the multiplicities here could be 0, that is, the V^\lambda  may not even appear. So think of the partitions now as being a superset of parameterising objects. For n=2, it is well known that L_2(V)\cong A^2V, that is, the symmetric square vanishes (just think of what the Lie bracket does here!).

Well I came across a beautiful result of Kraśkiewicz and Weyman that yields the answer. Consider standard tableaux of type \lambda. As an example, consider the partition (2,2,1,1) of the number 6. Then there are 9 possible standard tableaux:

tableauxA descent is an entry i of a standard tableaux such that i+1 appears in a row lower than it. So for example, 2, 4, and 5 are descents of the first tableau above. The major index of a standard tableaux is the sum of its descents. So for our nine tableaux we have major indices 11, 10, 9, 13, 9, 8, 12, 7, 11 accordingly. Then the magic number for the multiplicity of a V^\lambda is the number of standard tableaux of type \lambda that have major index congruent to 1 modulo n. In our example, there are two such young tableaux, the 4th and 8th one. So V^\lambda has multiplicity 2.

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