An algorithm for Berenstein-Kazhdan decoration functions and trails for classical Lie algebras

TL;DR

This paper presents an algorithm to explicitly compute Berenstein-Kazhdan decoration functions and trails for classical Lie algebras, enhancing the understanding of geometric crystal structures and their polyhedral realizations.

Contribution

It introduces a new algorithm for calculating summands of decoration functions for arbitrary reduced words in classical Lie types, including G2.

Findings

Algorithm successfully computes summands for classical types A, B, C, D.

Complete calculation of decoration functions possible for these types.

Algorithm verified for type G2, extending its applicability.

Abstract

For a simply connected connected simple algebraic group $G$, it is known that a variety $B_{w_0}^-:=B^-\cap U\overline{w_0}U$ has a geometric crystal structure with a positive structure $\theta^-_{\mathbf{i}}:(\mathbb{C}^{\times})^{l(w_0)}\rightarrow B_{w_0}^-$ for each reduced word $\mathbf{i}$ of the longest element $w_0$ of Weyl group. A rational function $\Phi^h_{BK}=\sum_{i\in I}\Delta_{w_0\Lambda_i,s_i\Lambda_i}$ on $B_{w_0}^-$ is called a half-potential, where $\Delta_{w_0\Lambda_i,s_i\Lambda_i}$ is a generalized minor. Computing $\Phi^h_{BK}\circ \theta^-_{\mathbf{i}}$ explicitly, we get an explicit form of string cone or polyhedral realization of $B(\infty)$ for the finite dimensional simple Lie algebra $\mathfrak{g}={\rm Lie}(G)$. In this paper, for an arbitrary reduced word $\mathbf{i}$, we give an algorithm to compute the summand $\Delta_{w_0\Lambda_i,s_i\Lambda_i}\circ \theta^-_{\mathbf{i}}$ of $\Phi^h_{BK}\circ \theta^-_{\mathbf{i}}$ in the case $i\in I$ satisfies that for any weight $\mu$ of $V(-w_0\Lambda_i)$ and $t\in I$, it holds $\langle h_t,\mu \rangle\in\{2,1,0,-1,-2\}$. In particular, if $\mathfrak{g}$ is of type ${\rm A}_n$, ${\rm B}_n$, ${\rm C}_n$ or ${\rm D}_n$ then all $i\in I$ satisfy this condition so that one can completely calculate $\Phi^h_{BK}\circ \theta^-_{\mathbf{i}}$. We will also prove that our algorithm works in the case $\mathfrak{g}$ is of type ${\rm G}_2$.