Let $G=GL(n,\mathbb{C})$ and let $U\subset G$ be a maximal unipotent subgroup. (For example,assume that U is the set of upper triangular matrices with ones in the diagonal.) Now let $X=M\_{n}(\mathbb{C})$ be the space of $n$ by $n$ complex matrices and consider the usual action of $U$ on $X$ given by left multiplication. Is it possible to describe the space $\mathbb{C}[X]^{U}$ explicitly using generators and relations? In other words, is there an explicit realization of the categorial quotient of $X$ by $U$ as an affine algebraic variety?



Invariants of the maximal unipotent subgroup of GL(n) acting on the space of n by n matrices
Here is a coordinate-free description of the affine hull of the quotient of $G=\text{Aut}\_k(V)$ by the right action of the unipotent radical $U$ of a Borel subgroup $B$. The group $B$ is the stabilizer of a flag of $k$-linear subspaces, $$\{0\} = F^0 \subsetneqq F^1 \subsetneqq \dots \subsetneqq F^r \subsetneqq F^n = V.$$ For $r=1,\dots,n-1$, denote by $H^r$ the affine $k$-space with underlying $k$-vector space $\text{Hom}\_k(\bigwedge^r F^r,\bigwedge^r V)$. Denote by $Z^r\subset H^r$ the affine cone over the Grassmannian $\text{Grass}(r,V)$, i.e., the homogeneous, Zariski closed subset parameterizing all linear maps of the form $\bigwedge^r T^r$ for a $k$-linear morphism $$T^r:F^r \to V.$$ The defining ideal of $Z^r$ in the (polynomial) coordinate ring $k[H^r]$ is generated by explicit Plücker quadratic relations, cf. Griffiths-Harris (for instance). Denote by $Z^n$ the open complement of the origin in the $1$-dimensional affine space $\text{Hom}\_k(\bigwedge^n F^n, \bigwedge^n V) =\text{End}\_k(\bigwedge^n V)$. 


Inside $Z^1 \times\_k Z^2 \times\_k \dots \times\_k Z^{n-1}$, let $Z'$ be the affine cone over the flag variety $\text{Flag}(1,2,\dots,n-1;V)$. This is cut out by explicit equations similar to the Plücker relations. For instance, for every $r=2,\dots,n-1$, one relation on $(z^1,\dots,z^{n-1})$ is that $z^1\wedge z^r$ equals $0$ as an element in $\text{Hom}\_k(\bigwedge^1F^1\otimes\_k \bigwedge^r F^r,\bigwedge^{r+1}V)$. I do realize that you want an explicit list of relations, and I am not quite giving these. However, you can quickly find the explicit relations by searching for "Cox ring" and "flag variety". 


Anyway, the affine hull of the quotient $G/U$ of the right action of $U$ on $G$ is $Z = Z'\times Z^n$. The quotient morphism is $$q: G \to Z, \ \ q(T) = \left( \bigwedge^1(T|\_{F^1}), \bigwedge^2(T|\_{F^2}),\dots, \bigwedge^{n-1}(T|\_{F^{n-1}}),\bigwedge^n(T|\_{F^n}) \right).$$
Although $Z$ is affine, the morphism $q$ is *not* surjective. In particular, an element $(z^1,\dots,z^{n-1},z^n)$ is in the image if and only if $z^r$ is nonzero for every $r=1,\dots,n-1$. This is the point I was making in my comment above.


**Edit.** I checked a textbook on the combinatorics of these kinds of commutative rings.


Combinatorial Commutative Algebra   

Ezra Miller, Bernd Sturmfels   

Graduate Texts in Mathematics (Book 227), 420 pages.   

Springer.   

ISBN-10: 0387237070   



On p. 275, Definition 14.2, they define the ring $k[Z']$ to be the **Plücker algebra**. It is generated by the linear bases for the dual vector spaces of $H^1,\dots,H^{n-1}$, which total $2^n-2$ variables. The ideal of relations is generated in degree $2$. Most of Chapter 14 of that book is devoted to describing these quadratic relations.


