Формулы характера операды пары согласованных скобок и бигамильтоновой операды
We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the SL2 group in these spaces.
Recall that the series of characters of symmetric groups form a symmetric function. SL2 grading gives an additional parameter to these functions. First, we show how Koszul duality implies the relation for symmetric functions. Namely, the symmetric functions corresponding to the characters of Koszul dual operads should be inverse to each other (up to minor correction of signs) with respect to the plethystic composition. Second, we present a formula for the inverse symmetric function generalizing the Moebius inversion formula.
We proved that the operad of two compatible Lie brackets is Koszul. The koszul dual operad has trivial action of symmetric groups and, therefore, it is easy to write the corresponding symmetric function. Then we use our inversion formula and simplify the result in order to find particular characters for the action of symmetric group.
I insist that the method we suggest is quite general and may be applied in many other situations.