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

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).