The product of two Schur functions can be decomposed into a linear combination of other Schur functions according to the Littlewood-Richardson rule. This is also how the irreducible representations in the tensor product of two finite-dimensional representations of the general linear group are calculated, which is how I originally ended up with this problem.
The integer partitions of a positive integer N form a lattice with respect to the dominance (or majorization) order of partitions. I.e., the infimum (or meet) and the supremum (or join) of any pair of partitions of N is also a partition of N.
I noticed that the set of partitions that correspond to the Schur functions in the decomposition of the product of any two Schur functions, seem to form a sublattice of the above lattice. The infimum and supremum of any pair of partitions in the product are also in the product, or at least it was true for any specific example I checked.
Is there some deeper reason for why this is so? Could someone give me pointers for how could I go about showing that it is true for the product of any two Schur functions? It doesn't seem to follow in a straightforward manner from the Littlewood-Richardson algorithm, and I don't know much about the theory of Schur functions.