Cluster monomials in $\mathbb{C}[G{L}_n/N]$, 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 $\mathbb{C}[G{L}_n/N]$. For we consider the {\em tableau basis} in $\mathbb{C}[G{L}_n/N]$. 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 $\mathbb{C}[G{L}_n/N]$ 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 $\mathbb{C}[G{L}_n/N]$, 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).

14 pages, 4 figures, extended abstract FPSAC15

References in corpus (1)

• Quantum cluster algebras and quantum nilpotent algebras