Hives and the fibres of the convolution morphism

TL;DR

This paper establishes a bijection between combinatorial hives and geometric components of affine Grassmannian fibres, linking combinatorics with geometric representation theory and proposing a conjectural extension involving the octahedron recurrence.

Contribution

It provides a simple bijection between hives and fibre components, offering a new geometric interpretation of combinatorial objects in representation theory.

Findings

Bijection between hives and fibre components established

Description of individual fibre components provided

Conjectural generalization involving octahedron recurrence proposed

Abstract

By the geometric Satake correspondence, the number of components of certain fibres of the affine Grassmannian convolution morphism equals the tensor product multiplicity for representations of the Langlands dual group. On the other hand, in the case of GL_n, combinatorial objects called hives also count tensor product multiplicities. The purpose of this paper is to give a simple bijection between hives and the components of these fibres. In particular, we give a description of the individual components. We also describe a conjectural generalization involving the octahedron recurrence.