Enumeration of graphs arising in invariant theory

Question

I've been working on a talk based on some stuff in Olver's "Classical Invariant Theory" book and have been wondering about a related graph enumeration problem.

Start with a triple $(n,v,e)$ of natural numbers. Take all $\mathbb{Q}$-linear combinations of directed graphs (allowing multiple edges, but no loops) with $v$ vertices, $e$ edges, and each vertex has at most $n$ edges going to it or coming from it. Now, take three relations (images scanned from Olver)

Rule 1:

Rule 2:

Rule 3:

Where the function $v$ with a vertex as subscript next to a graph means that graph multiplied by $n$ minus the number of edges attached to that vertex. (So, for instance, an isolated vertex gets multiplied by $n$)

Denote the space after quotienting by these relations by $V_{n,v,e}$. And so, in final form, my question:

What is $\dim V_{n,v,e}$? Or at least, can we find relatively effective upper bounds?

EDIT: Some clarifications. The colorings on the vertices are just to mark them in the pictures to keep track of where everything goes, the graphs are not marked themselves. Additionally, as Rule 2 is slightly unclear from the scan, the $v$ function is always the vertex not attached to the arrow in the configuration.

Answer 1

The number ${\rm dim}\ V_{n,v,e}$ is the number of linearly independent covariants of degree $v$ and weight $e$ of a binary form of degree $n$. This is the multiplicity of the irreducible module $Sym^k(\mathbb{C}^2)$ in the plethysm $Sym^v(Sym^n(\mathbb{C}^2))$, where $k=nv-2e$. There is formula for that involving counting integer partitions (Cayley-Sylvester formula). The easiest way to derive it that I know of is by computing the character which is a Gaussian polynomial. I think the book by Mukai does that. You can also look up http://arxiv.org/abs/math/0110224

Answer 2

I've been fascinated by this and never got to the bottom of it. I hesitate to offer this as a reply but I think this question deserves some response. I apologise if you already know this.

As I understand it (and please correct me if I go wrong) the fundamental problem introduced back in Chapter 2 is to understand the space of covariants of given degree $n$, order $p$, weight $w$ (with relation (2.33)). In the language of representations we take the $p$-th symmetric power of the $n$-th symmetric power of the defining representation of $GL(2)$. Then we take the isotypic subspace corresponding to $\det^{-w}$. Judging by the table on page 40 and the accompanying discussion the dimensions of these spaces are difficult to compute. My understanding is that even with current computers it is still difficult.

Each graph gives a covariant by some process I find obscure and the the graphical relations correspond to linear relations. By the first and second fundamental theorems the vector spaces you ask about are identified with the spaces of covariants (although the names of the parameters seem to have been changed).

The dimensions of the spaces of covariants are normally expressed in terms of Hilbert series and these can in principle be calculated (using Molien's theorem). For each $n$ this is a rational function. These are known for $n=1,2,3$ but rapidly become unwieldy and uninformative. It would seem plausible that the asymptotics of the Hilbert series can be calculated without finding the rational function.

Again, I expect I am simply rephrasing the question, not answering it.