5 research outputs found
Noncommutative Schur polynomials and the crystal limit of the U_q sl(2)-vertex model
Starting from the Verma module of U_q sl(2) we consider the evaluation module
for affine U_q sl(2) and discuss its crystal limit (q=0). There exists an
associated integrable statistical mechanics model on a square lattice defined
in terms of vertex configurations. Its transfer matrix is the generating
function for noncommutative complete symmetric polynomials in the generators of
the affine plactic algebra, an extension of the finite plactic algebra first
discussed by Lascoux and Sch\"{u}tzenberger. The corresponding noncommutative
elementary symmetric polynomials were recently shown to be generated by the
transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin
and Kitanine. Here we establish that both generating functions satisfy Baxter's
TQ-equation in the crystal limit by tying them to special U_q sl(2) solutions
of the Yang-Baxter equation. The TQ-equation amounts to the well-known
Jacobi-Trudy formula leading naturally to the definition of noncommutative
Schur polynomials. The latter can be employed to define a ring which has
applications in conformal field theory and enumerative geometry: it is
isomorphic to the fusion ring of the sl(n)_k -WZNW model whose structure
constants are the dimensions of spaces of generalized theta-functions over the
Riemann sphere with three punctures.Comment: 24 pages, 6 figures; v2: several typos fixe
On the uniqueness of promotion operators on tensor products of type A crystals
The affine Dynkin diagram of type has a cyclic symmetry. The
analogue of this Dynkin diagram automorphism on the level of crystals is called
a promotion operator. In this paper we show that the only irreducible type
crystals which admit a promotion operator are the highest weight crystals
indexed by rectangles. In addition we prove that on the tensor product of two
type crystals labeled by rectangles, there is a single connected
promotion operator. We conjecture this to be true for an arbitrary number of
tensor factors. Our results are in agreement with Kashiwara's conjecture that
all `good' affine crystals are tensor products of Kirillov-Reshetikhin
crystals.Comment: 31 pages; 8 figure