e-Positivity of vertical strip LLT polynomials

D'Adderio, Michele

doi:10.1016/j.jcta.2020.105212

Cited by 8 publications

(1 citation statement)

References 9 publications

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“…The case and follows from the proof of the Delta theorem and the e -positivity on vertical strip polynomials proved in [D’A20].…”

Section: Discussionmentioning

confidence: 99%

Delta and Theta Operator Expansions

Iraci,

Romero

2024

Forum of Mathematics, Sigma

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We give an elementary symmetric function expansion for the expressions $M\Delta _{m_\gamma e_1}\Pi e_\lambda ^{\ast }$ and $M\Delta _{m_\gamma e_1}\Pi s_\lambda ^{\ast }$ when $t=1$ in terms of what we call $\gamma $ -parking functions and lattice $\gamma $ -parking functions. Here, $\Delta _F$ and $\Pi $ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$ . Our main results, in turn, give an elementary basis expansion at $t=1$ for symmetric functions of the form $M \Delta _{Fe_1} \Theta _{G} J$ whenever F is expanded in terms of monomials, G is expanded in terms of the elementary basis, and J is expanded in terms of the modified elementary basis $\{\Pi e_\lambda ^\ast \}_\lambda $ . Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an e-positivity conjecture for when t is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.

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“…The case and follows from the proof of the Delta theorem and the e -positivity on vertical strip polynomials proved in [D’A20].…”

Section: Discussionmentioning

confidence: 99%

Delta and Theta Operator Expansions

Iraci,

Romero

2024

Forum of Mathematics, Sigma

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LLT polynomials, elementary symmetric functions and melting lollipops

Alexandersson

2020

J Algebr Comb

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We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Shareshian-Wachs conjecture previously studied by Panova and the author in 2018. We show that the conjecture for unicellular LLT polynomials implies a similar formula for vertical-strip LLT polynomials.We prove positivity in the elementary basis in for the class of graphs called "melting lollipops" previously considered by Huh, Nam and Yoo. This is done by proving a curious relationship between a generalization of charge and orientations of unit-interval graphs.We also provide short bijective proofs of Lee's three-term recurrences for unicellular LLT polynomials and we show that these recurrences are enough to generate all unicellular LLT polynomials associated with abelian area sequences.

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A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

Alexandersson

Sulzgruber

2022

Advances in Mathematics

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e-Positivity of vertical strip LLT polynomials

Cited by 8 publications

References 9 publications

Delta and Theta Operator Expansions

Delta and Theta Operator Expansions

LLT polynomials, elementary symmetric functions and melting lollipops

A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

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