Is it possible to classify the indecomposable representations of the wild quiver with one vertex and two arrows using infinite sets of parameters?

Question

I have read several other questions about wild representation type but may I ask..what actually can be done and what have been done (such as partial results) about classifying indecomposable representations of the quiver $\mathbb{F_2}$?

Especially is it possible to find infinite (perhaps even countable) sets of continuous and discrete parameters that can be used to classify its representation of a certain given dimension perhaps with some exceptions indexed by a known set? If this can be done than indecomposable representations of the quiver of one vertex and two arrows (which we call $\mathbb{F_2}$ from now on) will be classified.

Would you please provide some partial results (and especially recent ones) and some ideas about how this problem might be solved? Thank you very much!

Answer 1

The representation theory of the quiver with one vertex and two loops (that is, the problem of classifying finite dimensional representations of the free algebra $\mathbb{C} \langle x,y \rangle$ upto isomorphism) is 'in principle' accessible via the tools of geometric invariant theory.

First one fixes the dimension $n$ of the representation. Then the problem is trying to classify pairs of $n \times n$ matrices upto simultaneous conjugation, that is the orbits of $M_n(\mathbb{C}) \times M_n(\mathbb{C})$ under this action by $GL_n$.

The quotient variety $X_n = M_n(\mathbb{C}) \times M_n(\mathbb{C})/GL_n$ classifies the closed orbits which are the isomorphism classes of semi-simple representations. It is of dimension $n^2+1$ and its coordinate ring is generated by traces in words on $x$ and $y$. It can be stratified by locally closed smooth strata and the etale local structure in points $x \in S_{\tau}$ of such strata are given by similar quotient varieties of 'easier' (that is, lower dimensional) quiver-representations $(Q_{\tau},\alpha_{\tau})$.

To classify further all $n$-dimensional representations having as its Jordan-Holder semi-simplification the given point $x \in S_{\tau}$ amounts to studying the orbits in the nullcone of the quiver-setting $(Q_{\tau},\alpha_{\tau})$ and is best approached via the Hesselink-stratification, which again reduces to further quiver-representation problems (and vector-bundles over them).

An old account of this program is given in "Orbits of matrix tuples". Seminaires Congres, Societe Mathematique de France, 2:245–261, 1997 here's the preprint version : http://win.ua.ac.be/~lebruyn/LeBruyn1995c.pdf

When i said 'in principle' i mean that this program is only carried out in full detail for $n \leq 4$, and here's why:

Even for the generic stratum, corresponding to isomorphism classes of simple $n$-dimensional representations it is not known whether it is a rational variety (that is, whether one can parametrize simple orbits by a set of independent variables) unless $n \leq 4$ (by results of Procesi and Formanek).

For $n=5$ or more general for $n$ dividing $420 = 2^2.3.5.7$ we only know that the variety is stably rational and beyond those $n$ we do not know a thing. This is in an even older preprint: "Stable rationality of certain PGL(n) quotients". Inventiones Mathematicae, 104:179–199, 1991 http://win.ua.ac.be/~lebruyn/LeBruyn1989b.pdf

In recent years significant progress has been made in the study of moduli spaces of semi-stable quiver representations, look for instance on the arXiv for work by Markus Reineke. Aidan Schofield proved the remarkable fact that these moduli spaces are birational to the variety of tuples of $n \times n$ matrices under simultaneous conjugation where $n$ is the greatest common divisor of the components of the dimension vector, so eventually in all wild quiver-problems one bangs ones head against this most classical of all wild problems.