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Two character formulas for spaces of coinvariants
We consider spaces of coinvariants with respect to two kinds of
ideals of the enveloping algebra U(sl_2\otimes\C[t]). The first one is
generated by , and the second one is generated by where are fixed generic polynomials. (We also
treat a generalization of the latter.) Using a method developed in our previous
paper, we give new fermionic formulas for their Hilbert polynomials in terms of
the level-restricted Kostka polynomials and -multinomial symbols. As a
byproduct, we obtain a fermionic formula for the fusion product of
-modules with rectangular highest weights, generalizing a known result
for symmetric (or anti-symmetric) tensors.Comment: LaTeX, 22 pages; very minor change
Finite type modules and Bethe Ansatz for quantum toroidal gl(1)
We study highest weight representations of the Borel subalgebra of the
quantum toroidal gl(1) algebra with finite-dimensional weight spaces. In
particular, we develop the q-character theory for such modules. We introduce
and study the subcategory of `finite type' modules. By definition, a module
over the Borel subalgebra is finite type if the Cartan like current \psi^+(z)
has a finite number of eigenvalues, even though the module itself can be
infinite dimensional.
We use our results to diagonalize the transfer matrix T_{V,W}(u;p) analogous
to those of the six vertex model. In our setting T_{V,W}(u;p) acts in a tensor
product W of Fock spaces and V is a highest weight module over the Borel
subalgebra of quantum toroidal gl(1) with finite-dimensional weight spaces.
Namely we show that for a special choice of finite type modules the
corresponding transfer matrices, Q(u;p) and T(u;p), are polynomials in u and
satisfy a two-term TQ relation. We use this relation to prove the Bethe Ansatz
equation for the zeroes of the eigenvalues of Q(u;p). Then we show that the
eigenvalues of T_{V,W}(u;p) are given by an appropriate substitution of
eigenvalues of Q(u;p) into the q-character of V.Comment: Latex 42 page
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