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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?

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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
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1 Answer

Operads that are quotients of the associative operad (which I believe your updated question is aimed at) are, in characteristic zero, fairly extensively studied since the 1950s or so, under the somewhat misleading for geometers name "varieties of associative algebras". (One of the first papers that sort of created this field of study is, to the best of my knowledge, Specht's "Gesetze in Ringen", http://www.ams.org/mathscinet-getitem?mr=35274.) The main bit of knowledge one needs to translate papers on varieties of algebras into the operadic language are

"variety of associative algebras" = "quotient of the associative operad"

and

"codimensions of a variety"="dimensions of components of an operad".

Armed with this knowledge, have a look at recent surveys, for example the book of Giambruno and Zaicev (Google Books link: http://tinyurl.com/giambrunozaicev), and you will see many hints on how to approach this kind of questions. In all fairness, the main question people in this area are interested in is asymptotic growth rather than exact formulas, but on the way lots of things are done, including some representation-theoretic computations, so you might find what you need.

P.S. Feel free to e-mail me if you are interested in some particular operads, maybe I happen to know precisely what you want to find out.

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