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Medientyp:
E-Artikel
Titel:
LLT polynomials, elementary symmetric functions and melting lollipops
Beteiligte:
Alexandersson, Per
Erschienen:
Springer Science and Business Media LLC, 2021
Erschienen in:
Journal of Algebraic Combinatorics, 53 (2021) 2, Seite 299-325
Sprache:
Englisch
DOI:
10.1007/s10801-019-00929-z
ISSN:
0925-9899;
1572-9192
Entstehung:
Anmerkungen:
Beschreibung:
AbstractWe 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 symmetric basis 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.