An algorithm for Berenstein-Kazhdan decoration functions and trails for minuscule representations

TL;DR

This paper presents an explicit algorithm to compute Berenstein-Kazhdan decoration functions and trails for minuscule representations, providing a combinatorial tool that resembles crystal graphs, applicable to various types including A_n and G_2.

Contribution

The paper introduces a novel algorithm for explicit computation of decoration functions in minuscule representations, extending to non-minuscule cases like G_2.

Findings

The algorithm effectively computes decoration graphs for minuscule representations.

It generalizes to all i in type A_n and works for some non-minuscule cases like G_2.

The decoration graph shares properties with crystal graphs of minuscule representations.

Abstract

For a simply connected connected simple algebraic group $G$, a cell $B_{w_0}^-=B^-\cap U\overline{w_0}U$ is a geometric crystal with a positive structure $\theta_{\textbf{i}}^-:(\mathbb{C}^{\times})^{l(w_0)}\rightarrow B_{w_0}^-$. Applying the tropicalization functor to a rational function $\Phi^h_{BK}=\sum_{i\in I}\Delta_{w_0\Lambda_i,s_i\Lambda_i}$ called the half decoration on $B_{w_0}^-$, one can realize the crystal $B(\infty)$ in $\mathbb{Z}^{l(w_0)}$. By computing $\Phi^h_{BK}$, we get an explicit form of $B(\infty)$ in $\mathbb{Z}^{l(w_0)}$. In this paper, we give an algorithm to compute $\Delta_{w_0\Lambda_i,s_i\Lambda_i}\circ \theta_{\textbf{i}}^-$ explicitly for $i\in I$ such that $V(\Lambda_i)$ is a minuscule representation of $\mathfrak{g}={\rm Lie}(G)$. In particular, the algorithm works for all $i\in I$ if $\mathfrak{g}$ is of type ${\rm A}_n$. The algorithm computes a directed graph $DG$, called a decoration graph, whose vertices are labelled by all monomials in $\Delta_{w_0\Lambda_i,s_i\Lambda_i}\circ \theta_{\textbf{i}}^-(t_1,\cdots,t_{l(w_0)})$. The decoration graph has some properties similar to crystal graphs of minuscule representations. We also verify that the algorithm works in some other cases, for example, the case $\mathfrak{g}$ is of type ${\rm G}_2$ though $V(\Lambda_i)$ is non-minuscule.