Mathematics Colloquia and Seminars

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We define cylindric generalizations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux, which are periodic continuations of ordinary skew tableaux, employing an integrable statistical lattice model on non-intersecting paths. We show that the cylindric Macdonald functions appear in the coproduct of a commutative Frobenius algebra, which can be interpreted as a one-parameter deformation of the sl(n) Verlinde algebra, i.e. the structure constants of the Frobenius algebra are polynomials in a variable t whose constant terms are the Wess-Zumino-Novikov-Witten fusion coefficients. The latter are known to coincide with dimensions of moduli spaces of generalized theta-functions and multiplicities of tilting modules of quantum groups at roots of unity. Alternatively, the deformed Verlinde algebra can be realized 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 on a highest weight vector, one obtains Lusztig's canonical basis.