Alladi Ramakrishnan Hall

Schur expansions of multipartite partition generating functions

Amritanshu Prasad

IMSc

The multipartite partition function counts the number of ways of writing an integer vector with non-negative entries as a sum of vectors of the same kind. Another variant counts the number of ways of writing an integer vector with non-negative entries as a sum of vectors of the same kind, but with distinct summands. The study of their generating functions can be traced back to Major P. A. Macmahon's book Combinatory Analysis, vol.2, 1916.

The multivariate generating functions of these combinatorial functions are symmetric in their variables. We prove the surprising result that these symmetric functions have non-negative integer coefficients when expanded in the basis of polynomials. We give an interpretation of the expansion coefficients in terms of restriction of representations of GL(n) to its subgroup of permutation matrices.