Suppose I have a symmetric operad $\mathcal{P}$ defined over $\text{Vect}_{\mathbb{K}}$ with generators and relations in degrees at most $l$. Now suppose I already know $\mathcal{P}(k)$ as an $S_k$-representation for $0 \leq k \leq l$.

How do I determine $\mathcal{P}(l+1)$ from $\mathcal{P}(0), \mathcal{P}(1), \ldots, \mathcal{P}(l)$ ? More generally, how do I determine $\mathcal{P}(k)$ for some large $k$?

My current technique is to find vector space generators for each grade of the operadic ideal of relations by building trees in which some relation appears. Then I use linear algebra and Schur-Weyl duality to determine each grade of the ideal as a representation. This technique is reliable for small cases, but the number of trees grows large and this generating set is usually quite redundant.

I am hoping (perhaps naively) for some combinatorial rule in terms of Young diagrams which can answer the question explicitly.

Edit: Vladimir Dotsenko suggests that I limit the scope, so here's a (more reasonable) question which is also interesting:

Assume further that $\mathcal{P}$ is generated by a single associative binary operation. Now can we determine $\mathcal{P}(k)$ from an initial segment using the combinatorics of Young diagrams?

It seems like the Littlewood-Richardson rule might be of some help here. There also seems to be some plethysm going on. Perhaps these tools are sufficient?

• In this generality, nothing much better can be done, I think. If you are hoping for some suggestions, you might want to narrow things down a bit. – Vladimir Dotsenko Jun 13 '12 at 7:25
• @Vladimir Dostenko Good to know, thanks. I have added a smaller question which seems more likely to have an answer. – John Wiltshire-Gordon Jun 13 '12 at 16:10
• In your new setup, do you assume characteristic zero? – Vladimir Dotsenko Jun 14 '12 at 3:19
• @Vladimir Dostenko Yes. – John Wiltshire-Gordon Jun 14 '12 at 3:58
• @Vladimir Dotsenko Sorry I've been misspelling your name. – John Wiltshire-Gordon Jun 14 '12 at 4:04