Growth Diagrams and Minuscule Polygon Configurations in the Affine Grassmannian

TL;DR

This paper introduces affine growth diagrams related to the affine Grassmannian, establishing a bijection with convolution varieties for minuscule weights and connecting to classical combinatorial structures like Fomin growth diagrams and the RS-correspondence.

Contribution

It defines affine growth diagrams, proves a bijection with components of convolution varieties in the affine Grassmannian, and provides a combinatorial construction linking to classical combinatorics.

Findings

Affine growth diagrams correspond bijectively to components of convolution varieties.

The construction generalizes classical Fomin growth diagrams.

Connections to the RS-correspondence are established.

Abstract

We define affine growth diagrams consisting of $GL_m$ dominant weights that label the vertices of a staircase-shaped grid. These are also called cylindrical growth diagrams as defined by Speyer and White in the case of partitions. The weights labelling each adjacent pair of vertices differ by a vertical strip and the weights around each unit square satisfy a local condition that appeared in van Leeuwen's work on the Littelmann path model for crystals. We prove two main results. For a sequence of minuscule weights $\vec\lambda=(\lambda^1,\ldots,\lambda^n)$ let Poly$(\vec\lambda)$ denote the configuration space of $n$-tuples of points $(g_1,\ldots,g_n)$ in the affine Grassmannian such that the weight-valued distances satisfy $d(g_i,g_{i+1})=\lambda^i$. This is the convolution variety arising in the geometric Satake correspondence. We show that for a generic point $(g_1,\ldots,g_n)$ of a component the distances $d(g_i,g_j)$ form an affine growth diagram and that this gives a bijection between components of Poly$(\vec\lambda)$ and affine growth diagrams of type $\vec\lambda$. The main tool used in the proof is the Knutson--Tao hive. In the second part, we give a purely combinatorial construction of affine growth diagrams from natural number entries by applying Greene's theorem to certain subrectangles of the staircase. From this construction it follows that affine growth diagrams contain the classical Fomin growth diagrams and realize the RS-correspondence when $\vec\lambda=(\omega_1,\ldots,\omega_1,\omega_1^*,\ldots,\omega_1^*)$.