# Cylindric versions of specialised Macdonald functions and a deformed
Verlinde algebra

Research paper by **Christian Korff**

Indexed on: **06 Sep '12**Published on: **06 Sep '12**Published in: **Mathematical Physics**

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#### 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.