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
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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