Invariants of matrix pairs over discrete valuation rings and Littlewood–Richardson fillings

Let M and N be two r×r matrices of full rank over a discrete valuation ring R with residue field of characteristic zero. Let P,Q and T be invertible r×r matrices over R. It is shown that the orbit of the pair (M,N) under the action (M,N)↦(PMQ-1,QNT-1) possesses a discrete invariant in the form of Littlewood–Richardson fillings of the skew shape λ/μ with content ν, where μ is the partition of orders of invariant factors of M, ν is the partition associated to N, and λ the partition of the product MN. That is, we may interpret Littlewood–Richardson fillings as a natural invariant of matrix pairs. This result generalizes invariant factors of a single matrix under equivalence, and is a converse of the construction in Appleby (1999) [1], where Littlewood–Richardson fillings were used to construct matrices with prescribed invariants. We also construct an example, however, of two matrix pairs that are not equivalent but still have the same Littlewood–Richardson filling. The filling associated to an orbit is determined by special quotients of determinants of a matrix in the orbit of the pair.