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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.

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    $\begingroup$ It is not true that $G = U_- B$ (unless $n = 1$, I guess). The set $U_- B$ is dense and open, but it is only the "big cell" of the Bruhat decomposition, and there are other (lower-dimensional) cells that are needed to get all of $G$. However, you are right that there is an embedding $\mathbb{C}[G/U_-] \hookrightarrow \mathbb{C}[B]$. (In your example, don't you also need the functions $g_{22}$, $g_{23}$, and $g_{33}$?) $\endgroup$ – Dave Witte Morris May 3 '16 at 3:40
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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.

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  • $\begingroup$ thank you very much. Is the action of $T$ on $G/U^-$ given by the following formula $G/U^- \times T \to G/U^-$, $(gU^-, t) \mapsto gt'U^-$ ($t'$ satisfies $t'U^- = U^- t$)? Since we have an action $G/U^- \times T \to G/U^-$, there is a coaction $\mathbb{C}[G/U^-] \to \mathbb{C}[G/U^-] \otimes \mathbb{C}[T]$. What is the corresponding multigrading on $\mathbb{C}[G/U^-]$? $\endgroup$ – Jianrong Li May 3 '16 at 8:40
  • $\begingroup$ First: $t'=t$. Second: $f$ is homogeneous of degree $\lambda$ if the coaction sends $f$ to $f\otimes\lambda$. $\endgroup$ – Friedrich Knop May 3 '16 at 10:59
  • $\begingroup$ I have another question. How to write the map $\mathbb{C}[G/U_-] \hookrightarrow \mathbb{C}[B]$ explicitly? For example, let $G = GL_2$. Then we have $\mathbb{C}[G/U_-] = \mathbb{C}[g_{11}, g_{21}, g_{11}g_{22} - g_{12} g_{21}]$ and $\mathbb{C}[B] = \mathbb{C}[b_{11}, b_{12}, b_{22}]$. What are the images of $g_{11}, g_{21}, g_{11}g_{22} - g_{12} g_{21}$? $\endgroup$ – Jianrong Li May 3 '16 at 12:33
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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.

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