Embedding of bases: from the M(2,2k+1) to the M(3,4k+2-delta) models

A new quasi-particle basis of states is presented for all the irreducible modules of the M(3,p) models. It is formulated in terms of a combination of Virasoro modes and the modes of the field phi_{2,1}. This leads to a fermionic expression for particular combinations of irreducible M(3,p) characters, which turns out to be identical with the previously known formula. Quite remarkably, this new quasi-particle basis embodies a sort of embedding, at the level of bases, of the minimal models M(2,2k+1) into the M(3,4k+2-delta) ones, with 0 \leq delta \leq 3.Comment: corrected a typo in the title, 7 page

Schubert varieties and the fusion products

For each $A\in\N^n$ we define a Schubert variety $\sh_A$ as a closure of the \Slt(\C[t])-orbit in the projectivization of the fusion product $M^A$. We clarify the connection of the geometry of the Schubert varieties with an algebraic structure of $M^A$ as \slt\otimes\C[t] modules. In the case when all the entries of $A$ are different $\sh_A$ is smooth projective algebraic variety. We study its geometric properties: the Lie algebra of the vector fields, the coordinate ring, the cohomologies of the line bundles. We also prove, that the fusion products can be realized as the dual spaces of the sections of these bundles.Comment: 34 page

Two dimensional current algebras and affine fusion product

In this paper we study a family of commutative algebras generated by two infinite sets of generators. These algebras are parametrized by Young diagrams. We explain a connection of these algebras with the fusion product of integrable irreducible representations of the affine $sl_2$ Lie algebra. As an application we derive a fermionic formula for the character of the affine fusion product of two modules. These fusion products can be considered as a simplest example of the double affine Demazure modules.Comment: 22 page

Quasi-invariants of dihedral systems

A basis of quasi-invariant module over invariants is explicitly constructed for the two-dimensional Coxeter systems with arbitrary multiplicities. It is proved that this basis consists of $m$-harmonic polynomials, thus the earlier results of Veselov and the author for the case of constant multiplicity are generalized.Comment: 22 pages; a minor correction done; accepted by Mathematical Note

Homogeneous components in the moduli space of sheaves and Virasoro characters

The moduli space $\mathcal M(r,n)$ of framed torsion free sheaves on the projective plane with rank $r$ and second Chern class equal to $n$ has the natural action of the $(r+2)$-dimensional torus. In this paper, we look at the fixed point set of different one-dimensional subtori in this torus. We prove that in the homogeneous case the generating series of the numbers of the irreducible components has a beautiful decomposition into an infinite product. In the case of odd $r$ these infinite products coincide with certain Virasoro characters. We also propose a conjecture in a general quasihomogeneous case.Comment: Published version, 19 page