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

TL;DR

This paper introduces a linear algebra-based method to map pairs of modules in the affine Grassmannian over discrete valuation rings to combinatorial objects called hives, extending prior work on matrix pairs and MV polytopes.

Contribution

It provides an elementary, ring-agnostic construction of hives from module pairs, generalizing previous results and connecting to conjectures on Hermitian matrices.

Findings

Map from module pairs to hives using invariant factors

Applicable over any discrete valuation ring

Links to conjectured hive constructions from Hermitian matrices

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.