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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, anelement $\Delta_T$ of the tableau basis labeled by a semistandard Young tableau$T$ is the product of the flag minors corresponding to columns of $T$. Our mainresults state: (i) cluster monomials in $\mathbb C[GL_n/N]$ can be labeled bysemistandard Young tableaux such that any cluster monomial has the form$\Delta_T+$ lexicographically smaller terms; (ii) such labeling distinguish thecluster monomial; (iii) for any seed of the cluster algebra on $\mathbbC[GL_n/N]$, we define a cone in $\mathbf D(n)$ generated by tableaux whichlabel the cluster variables of the seed, then these cones form a simlicial fanin $\mathbf D(n)$ ($\mathbf D(n)$ is linear isomorphic to the Gelfand Tseitlincone).

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