It's slightly nicer to look at $M_n // U_-$ instead of $GL(n) // U_-$, since then we're looking at invariants inside a polynomial ring. Namely, the ring generated by all determinants that use any $k$ rows and the left $k$ columns (for all $k=1,\ldots,n$); there are $2^n-1$ choices of row set.

These Plücker coordinates form a SAGBI basis for this ring, making it easy to write down a basis for each $T\times T$-weight space, in bijection with the relevant Gel$'$fand-Cetlin patterns. (Specifically, there are $2^n-1$ patterns consisting of only $0,1$, which are in obvious bijection with the Plücker coordinates. Then any pattern is canonically (though not uniquely) a sum of these basic patterns, suggesting the corresponding monomial.) This is in Miller and Sturmfels' "Combinatorial Commutative Algebra", chapter 14.

It's slightly nicer to look at $M_n // U_-$ instead of $GL(n) // U_-$, since then we're looking at a subring of invariants inside a polynomial ring. Namely, the subring generated by all determinants that use any $k$ rows and the left $k$ columns (for all $k=1,\ldots,n$); there are $2^n-1$ choices of row set.

These Plücker coordinates form a SAGBI basis for this ring, making it easy to write down a basis for each $T\times T$-weight space, in bijection with the relevant Gel$'$fand-Cetlin patterns. (Specifically, there are $2^n-1$ patterns consisting of only $0,1$, which are in obvious bijection with the Plücker coordinates. Then any pattern is canonically (though not uniquely) a sum of these basic patterns, suggesting the corresponding monomial.) This is in Miller and Sturmfels' "Combinatorial Commutative Algebra", chapter 14.