We prove the polynomiality of the bigraded ring $J_{*,*}^{w, W}(F_4)$ of weak Jacobi forms for the root system $F_4$ which are invariant with respect to the corresponding Weyl group. This work is a continuation of the joint article with V.A. Gritsenko, where the structure of algebras of the weak Jacobi forms related to the root systems of $D_n$ type for $2\leqslant n \leqslant 8$ was studied.

The Thoma simplex Ω is an infinite-dimensional space, a kind of dual object to the infinite symmetric group. The z-measures are probability measures on Ω depending on three continuous parameters. One of them is the parameter of the Jack symmetric functions, and in the limit as it goes to 0, the z-measures turn into the Poisson–Dirichlet distributions. The definition of the z-measures is somewhat implicit. We show that the topological support of any nondegenerate z-measure is the whole space Ω.

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with certain finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. We discuss a universal additive topological invariant of V-manifolds, the universal Euler characteristic. It takes values in the ring freely generated (as a Z-module) by isomorphism classes of finite groups. We also consider the universal Euler characteristic on the class of locally closed equivariant unions of cells in equivariant CW-complexes. We show that it is a universal additive invariant satisfying a certain “induction relation.” We give Macdonald-type identities for the universal Euler characteristic for V-manifolds and for cell complexes of the described type.

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.