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Cylindric Versions of Specialised Macdonald Functions and a Deformed Verlinde Algebra

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We define cylindric versions of skew Macdonald functions P λ/μ (q, t) for the special cases q = 0 or t = 0. Fixing two integers n > 2 and k > 0 we shift the skew diagram λ/μ, 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 and associated weight functions. The resulting weighted sums over these cylindric tableaux are symmetric functions. They 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 \({U_{q}\widehat{\mathfrak{sl}}(n)}\) Kirillov-Reshetikhin module. 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 = t = 0, this Frobenius algebra is isomorphic to the \({\widehat{\mathfrak{sl}}(n)}\) Verlinde algebra at level k, i.e. the structure constants become the \({\widehat{\mathfrak{sl}}(n)_{k}}\) Wess-Zumino-Novikov-Witten fusion coefficients. Further motivation comes from exactly solvable lattice models in statistical mechanics: the cylindric Macdonald functions discussed here arise as partition functions of so-called vertex models obtained from solutions to the 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.

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    Christian Korff

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Korff, C. Cylindric Versions of Specialised Macdonald Functions and a Deformed Verlinde Algebra. Commun. Math. Phys. 318, 173–246 (2013). https://doi.org/10.1007/s00220-012-1630-9

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