Séminaire Lotharingien de Combinatoire, 80B.69 (2018), 12 pp.
Emily Sergel
A Parking Function Interpretation for ∇mn,1k
Abstract.
The modified Macdonald polynomials introduced by Garsia and Haiman
(1996) have many remarkable combinatorial properties. One such class
of properties involves applying the ∇ operator of Bergeron
and Garsia (1999) to basic symmetric functions. The first discovery of
this type was the Shuffle Conjecture of Haglund, Haiman, Loehr,
Remmel, and Ulyanov (2005), which relates the expression ∇
to parking functions. A refinement of this conjecture, called
the Compositional Shuffle Conjecture, was introduced by Haglund,
Morse, and Zabrocki (2012) and proved by Carlsson and Mellit (2015).
We give a symmetric function identity relating hook monomial symmetric
functions to the operators used in the Compositional Shuffle
Conjecture. This implies a parking function interpretation for nabla
of a hook monomial symmetric function, as well as LLT positivity. We
show that our identity is a q-analog of the expansion of a hook
monomial into complete homogeneous symmetric functions given by
Kulikauskas and Remmel (2006). We use this connection to conjecture a
model for expanding ∇mλ in this way when λ
is not a hook.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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