TITLE: Invariants of the maximal unipotent subgroup of GL(n) acting on the space of n by n matrices
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: Questions of this type can be approached from the direction of algebraic geometry (in a suitable generality) or from the direction of invariant theory involving algebraic group actions.   I'd point especially to Chapter 3 in the 1997 monograph by Frank D. Grosshans, Algebraic homogeneous spaces and invariant theory, Lect. Notes in Math. 1673, Springer.   He works in some generality throughout, taking $k$ to be any algebraically closed field of any characteristic and using "traditional" algebraic geometry language.  In particular, he looks at the case of general linear groups in $\S13$ of that chapter and describes (for either left or right action) the rings of invariants for $U$ on suitable algebras of polynomials.
Grosshans builds on older foundations for reductive group actions on varieties and resulting homogeneous spaces, going back to Hilbert, Mumford, and others.  His results rely a lot on combinatorics and are often more comprehensive than what you ask for.   On the other hand, he is fairly straightforward and careful about the details.  Though he works in arbitrary characteristic, his approach is not too different from that of the Russian school (Vinberg, Popov, Panyushev, ...) who typically work over an algebraically closed field of characteristic 0.  Along the way, Grosshans does provide a fairly detailed concrete picture of the quotients obtained, starting with the generators of rings of invariants.