Séminaire Lotharingien de Combinatoire, B79c (2019), 20 pp.

James Haglund, Brendon Rhoades and Mark Shimozono

Hall-Littlewood Expansions of Schur Delta Operators at t=0

Abstract. For any Schur function sν, the associated delta operator Δ'sν is a linear operator on the ring of symmetric functions which has the modified Macdonald polynomials as an eigenbasis. When ν = (1n-1) is a column of length n-1, the symmetric function Δ'en-1en appears in the Shuffle Theorem of Carlsson and Mellit. More generally, when ν = (1k-1) is any column the polynomial Δ'ek-1en is the symmetric function side of the Delta Conjecture of Haglund, Remmel, and Wilson. We give an expansion of ωΔ'sνen at t=0 in the dual Hall-Littlewood basis for any partition ν. The Delta Conjecture at t=0 was recently proven by Garsia, Haglund, Remmel, and Yoo; our methods give a new proof of this result. We give an algebraic interpretation of ωΔ'sνen at t=0 in terms of a Hom-space.

Received: February 1, 2018. Accepted: February 27, 2019. Final Version: February 28, 2019.

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