S. Loktev

Multi-Dimensional Weyl Modules and Symmetric Functions

The Weyl modules in the sense of V. Chari and A. Pressley ([CP]) over the current Lie algebra on an affine variety are studied. We show that local Weyl modules are finite-dimensional and generalize the tensor product decomposition theorem from [CP]. More explicit results are stated for currents on a non-singular affine variety of dimension d with coefficients in the Lie algebra slr. The Weyl modules with highest weights proportional to the vector representation one are related to the multi-dimensional analogs of harmonic functions. The dimensions of such local Weyl modules are calculated in the following cases. For d=1 we show that the dimensions are equal to powers of r. For d=2 we show that the dimensions are given by products of the higher Catalan numbers (the usual Catalan numbers for r=2).

Spaces of coinvariants and fusion product. I. From equivalence theorem to Kostka polynomials.

The fusion rule gives the dimensions of spaces of conformal blocks in the WZW theory. We prove a dimension formula similar to the fusion rulefor spaces of coinvariants of affine Lie algebras g^. An equivalence of filtered spaces is established between spaces of coinvariants of two objects: highest weight g^-modules and tensor products of finite-dimensional evaluation representations of g\otimes\C[t]. In the sl_2 case we prove that their associated graded spaces are isomorphic to the spaces of coinvariants of fusion products, and that their Hilbert polynomials are the level-restricted Kostka polynomials.

Combinatorics of the\(\widehat{\mathfrak{s}\mathfrak{l}}_2 \) spaces of coinvariants

AbstractWe consider two types of quotients of the integrable modules of

Klein Topological Field Theories from Group Representations

\widehat{\mathfrak{s}\mathfrak{l}}_2

Combinatorics of the

. These spaces of coinvariants have dimensions described in terms of the Verlinde algebra of levelk. We describe monomial bases for the spaces of coinvariants, which leads to a fermionic description of these spaces. Fork=1, we give explicit formulas for the characters. We also present recursion relations satisfied by the characters and the monomial bases.

\widehat{\germ{sl}}\sb 2

In this paper we prove an identity generalizing the Andrews-Gordon identity. We also discuss the meaning of our formula from the viewpoints of geometry of affine flag varieties and of geometry of polyhedra.

coinvariants: dual functional realization and recursion.

We show that any complex (respectively real) representation of finite group naturally generates a open-closed (respectively Klein) topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in this theory with the Frobenius{Schur indicator on the representation. We relate any complex simple Klein TFT to a real division ring.

Coinvariants for lattice VOAs and

Local Weyl modules over two-dimensional currents with values in Glr are deformed into spaces with bases related to parking functions. Using this construction, we (1) propose a simple proof that dimension of the space of diagonal coinvariants is not less than the number of parking functions; (2) describe the limits of Weyl modules in terms of semi-infinite forms and find the limits of characters; and (3) propose a lower bound and state a conjecture for dimensions of Weyl modules with arbitrary highest weights. Also we express characters of deformed Weyl modules in terms of ρ-parking functions and the Frobenius characteristic map.