Kernel function and quantum algebras

2010/02/12 by Feigin, B., Hoshino, A., Shibahara, J. +2 · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1002.2485

Abstract

We introduce an analogue K_n(x,z;q,t) of the Cauchy-type kernel function for the Macdonald polynomials, being constructed in the tensor product of the ring of symmetric functions and the commutative algebra A over the degenerate ℂ ℙ¹. We show that a certain restriction of K_n(x,z;q,t) with respect to the variable z is neatly described by the tableau sum formula of Macdonald polynomials. Next, we demonstrate that the integer level representation of the Ding-Iohara quantum algebra naturally produces the currents of the deformed W algebra. Then we remark that the K_n(x,z;q,t) emerges in the highest-to-highest correlation function of the deformed W algebra.

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