Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra

TL;DR

This paper introduces cylindric generalizations of skew Macdonald functions, linking them to quantum algebra structures, Verlinde algebras, and solvable lattice models, revealing new algebraic and combinatorial insights.

Contribution

It defines cylindric skew Macdonald functions as sums over cylindric skew tableaux and connects them to Frobenius algebras, quantum affine algebras, and statistical mechanics models.

Findings

Cylindric Macdonald functions form a basis for a Frobenius algebra related to quantum groups.

In the q=0 limit, the algebra recovers the sl(n) Verlinde algebra.

Cylindric functions correspond to partition functions of vertex models in statistical mechanics.

Abstract

We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an ordinary skew diagram of two partitions, viewed as a subset of the two-dimensional integer lattice, by the period vector (n,-k). Imposing a periodicity condition one defines cylindric skew tableaux as a map from the periodically continued skew diagram into the integers. The resulting cylindric Macdonald functions appear in the coproduct of a commutative Frobenius algebra, which is a particular quotient of the spherical Hecke algebra. We realise this Frobenius algebra as a commutative subalgebra in the endomorphisms over a Kirillov-Reshetikhin module of the quantum affine sl(n) algebra. Acting with special elements of this subalgebra, which are noncommutative analogues of Macdonald polynomials, on a highest weight vector, one obtains Lusztig's canonical basis. In the limit q=0, one recovers the sl(n) Verlinde algebra, i.e. the structure constants of the Frobenius algebra become the WZNW fusion coefficients which are known to be dimensions of moduli spaces of generalized theta-functions and multiplicities of tilting modules of quantum groups at roots of unity. Further motivation comes from exactly solvable lattice models in statistical mechanics: the cylindric Macdonald functions arise as partition functions of so-called vertex models obtained from solutions to the quantum Yang-Baxter equation. We show this by stating explicit bijections between cylindric tableaux and lattice configurations of non-intersecting paths. Using the algebraic Bethe ansatz the idempotents of the Frobenius algebra are computed.