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

Abstract: Let ${\mathcal O}$ be a discrete valuation ring with quotient field ${\cal K}$. The affine Grassmannian ${\cal G}r$ is the set of full-rank ${\mathcal O}$-modules contained in ${\cal K}^n$. Given $\Lambda \in {\cal G}r$, invariant factors $inv(\Lambda)=\lambda \in {\mathbb Z}^n$ stratify ${\cal G}r$. Left-multiplication by $GL_{n}({\cal K})$ stratifies ${\cal G}r \times {\cal G}r$ where $inv(N,\Lambda) = \mu$ if $(N,\Lambda)$ and $(I_{n} ,M)$ are in the same $GL_{n}({\cal K})$ orbit, and $inv(M) = \mu$. We present an elementary map from ${\cal G}r \times {\cal G}r$ to hives (in the sense of Knutson and Tao) of type $(\mu,\nu,\lambda)$ where $inv(N,\Lambda) = \mu$, $inv(N) = \nu$, and $inv(\Lambda) = \lambda$. Earlier work by the authors determined Littlewood-Richardson fillings from matrix pairs over certain rings ${\mathcal O}$, and later Kamnitzer utilized properties of MV polytopes to define a map from ${\cal G}r\times {\cal G}r$ to hives over ${\mathcal O} = {\mathbb C}[[t]]$. 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.