How to write $\mathbb{C}[G/U_-]=\oplus_{\lambda} V_{\lambda}$ explicitly?

Question

Let $G=GL_n$ and $U_-$ the set of all lower unipotent triangular matrices. Then by Gauss Decomposition, we have $G = U_-B$, where $B$ is the set of all upper triangular matrices. The group $U_-$ acts on $G$ by left multiplication. Since $G = U_-B$, we have an embedding $B \hookrightarrow U_-B/U_- = G/U_-$. Therefore $\mathbb{C}[G/U_-] \hookrightarrow \mathbb{C}[B]$. We have $\mathbb{C}[G/U_-] = \oplus_{\lambda} V_{\lambda}$, where $\lambda$ is a dominant weight and $V_{\lambda}$ is the highest weight module of $U(\mathfrak{g})$ with highest weight $\lambda$, where $\mathfrak{g}$ is the Lie algebra of $G$.

My question is: how to write $\mathbb{C}[G/U_-]=\oplus_{\lambda} V_{\lambda}$ explicitly?

For example, let $G = GL_3$. Then \begin{align} \mathbb{C}[G/U_-] = \mathbb{C}[g_{11}, g_{12}, g_{13}], \end{align} where $g_{11}, g_{12}, g_{13}$ are coordinate functions which sends a matrix $x$ to matrix coefficients $x_{11}, x_{12}, x_{13}$ respectively. How to write the $V_{\lambda}$ in \begin{align} \mathbb{C}[G/U_-] = \mathbb{C}[g_{11}, g_{12}, g_{13}]=\oplus_{\lambda} V_{\lambda} \end{align} explicitly? Thank you very much.

Answer 1

The subspace $V_\lambda$ is very easy to see. Since $U^-$ is normalized by the maximal torus $T$ there is an action of $T$ on $G/U^-$ on the right. This means that $\mathbb C[G/U^-]$ carries a multigrading and $V_\lambda$ is just one of the multihomogeneous pieces.

The problem is now that $\mathbb C[G/U^-]$ is very complicated as a ring. The case $G=SL(2)$ is deceiving. Here $\mathbb C[G/U^-]$ is indeed a polynomial ring in two variables and the $V_\lambda$ are the binary forms of degree $\lambda$.

In general, $\mathbb C[G/U^-]$ is generated by the coordinate functions on all fundamental representations. The relations can be shown to be all quadratic, a generalization of the Plücker relations. For example for $G=GL(n)$ the ring is generated by all "left justified" $m\times m$-minors for $m=1,\ldots,n$.

So your description of $\mathbb C[G/U^-]$ for $G=GL(3)$ is incorrect. In fact $G/U^-$ is $6$-dimensional. The right hand side is the affine cone over a partial flag variety namely the projective plane, hence affine $3$-space.

The description of $\mathbb C[G/U^-]$ as a ring is precisely the purpose of standard monomial theory. There, one constructs a set of monomials which forms a $\mathbb C$-basis and describes multiplication by straightening laws.

Answer 2

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.