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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 *s*_{ν}, 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 = (1^{n-1}) is a column of length *n*-1, the symmetric function
Δ'_{en-1}*e*_{n} appears in the Shuffle Theorem of Carlsson and Mellit.
More generally, when
ν = (1^{k-1}) is any column the polynomial Δ'_{ek-1}*e*_{n}
is the symmetric function side of the Delta Conjecture of
Haglund, Remmel, and Wilson.
We give an expansion of ωΔ'_{sν}*e*_{n} 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'_{sν}*e*_{n} 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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