Matrix pairs over valuation rings and ℝ-valued Littlewood–Richardson fillings

Abstract

In the authors' previous work [G. Appleby and T. Whitehead, Invariants of matrix pairs over discrete valuation rings and Littlewood–Richardson fillings, Linear Algebra Appl. 432 (2010), pp. 1277–1298], an explicit method was developed to associate a Littlewood–Richardson filling {k ij } of the skew shape λ/μ with content ν to a pair of square matrices (M, N) defined over a discrete valuation ring of equicharacteristic zero. These results are here significantly extended to include rings possessing a real-valued valuation, along with a new, real-valued extension of the concept of a Littlewood–Richardson filling which allows for some filling-parts of negative length. A previously known combinatorial bijection between Littlewood–Richardson fillings establishing the equality of Littlewood–Richardson coefficients is generalized to the real-valued case, and shown to hold for the fillings associated with the matrix case as well. These results are obtained by deriving some precise descriptions of the behaviour of real-valued Littlewood–Richardson fillings under continuous deformation of the parameters of the filling.