Hives Determined by Pairs in the Affine Grassmannian over Discrete Valuation Rings

Abstract

Let O be a discrete valuation ring with quotient field K. The affine Grassmannian Gr is the set of full-rank O-modules contained in K n . Given Λ ∈ Gr, invariant factors inv(Λ) = λ ∈ Z n stratify Gr. Left-multiplication by GL n (K) stratifies Gr × Gr where inv(N, Λ) = µ if (N, Λ) and (I n , M ) are in the same GL n (K) orbit, and inv(M ) = µ. We present an elementary map from Gr ×Gr to hives (in the sense of Knutson and Tao) of type (µ, ν, λ) where inv(N, Λ) = µ, inv(N ) = ν, and inv(Λ) = λ. Earlier work by the authors [3] determined Littlewood-Richardson fillings from matrix pairs over certain rings O, and later Kamnitzer [7] utilized properties of MV polytopes to define a map from Gr × Gr to hives over O = C[[t]]. Our proof uses only linear algebra methods over any discrete valuation ring, where hive entries are minima of sums of orders of invariant factors over certain submodules. Our map is analogous to a conjectured construction of hives from Hermitian matrix pairs due to Danilov and Koshevoy [5].