Cluster monomials in $\mathbb C[GL_n/N]$, a simplicial fan in the cone of semi-standard Young tableaux, and the Lusztig basis

TL;DR

This paper explores the structure of cluster monomials in the coordinate ring of $GL_n/N$, introducing a tableau basis and demonstrating a simplicial fan structure in the associated cone, linking combinatorics and algebra.

Contribution

It establishes a labeling of cluster monomials by semistandard Young tableaux and constructs a simplicial fan structure in the cone of tableaux, connecting cluster algebra and representation theory.

Findings

Cluster monomials are labeled by semistandard Young tableaux.

The labeling uniquely identifies cluster monomials.

Cones generated by cluster variables form a simplicial fan in the tableau cone.

Abstract

We study the cluster monomials and cluster complex in $\mathbb C[GL_n/N]$. For we consider the {\em tableau basis} in $\mathbb C[GL_n/N]$. Namely, an element $\Delta_T$ of the tableau basis labeled by a semistandard Young tableau $T$ is the product of the flag minors corresponding to columns of $T$. Our main results state: (i) cluster monomials in $\mathbb C[GL_n/N]$ can be labeled by semistandard Young tableaux such that any cluster monomial has the form $\Delta_T+$ lexicographically smaller terms; (ii) such labeling distinguish the cluster monomial; (iii) for any seed of the cluster algebra on $\mathbb C[GL_n/N]$, we define a cone in $\mathbf D(n)$ generated by tableaux which label the cluster variables of the seed, then these cones form a simlicial fan in $\mathbf D(n)$ ($\mathbf D(n)$ is linear isomorphic to the Gelfand Tseitlin cone).