LLT polynomials, elementary symmetric functions and melting lollipops

Abstract

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.

Triangular Diagonal Harmonics Conjectures

“…We would like to have an expansion of F τ of the form To simplify the presentation of further expressions, let consider F (1) τ := A τ ⊗ 1 + 0 α⊆τ C (1) α ⊗ s (α+1 ℓ(α) )/α , where C (1) α := s n , with n = |τ | − |α|. In other words, this is the contribution that corresponds to rules (1) and (2).…”

“…cit. for details, as well as [1,10,11,17] for more context on the whole subject. The E (n) τ (q, t; x) are symmetric in q and t, and "Schur positive" both as functions of q and t, and the x-variables.…”

A horizontal-strip LLT polynomial is determined by its weighted graph

“…If λ is a sequence of single cells, then the unicellular LLT polynomial G λ (x; q) can be expressed as a sum over arbitrary colourings of a unit interval graph Γ(λ) associated to λ. Huh, Nam, and Yoo [HNY20] proved a combinatorial Schur expansion of G λ (x; q) whenever Γ(λ) is a "melting lollipop" and Alexandersson conjectured [Ale21] and then proved with Sulzgruber [AS22] a combinatorial elementary symmetric function expansion of G λ (x; q+1) in terms of acyclic orientations of Γ(λ). An equality of unicellular LLT polynomials G λ (x; q) = G µ (x; q) is equivalent [CM18] to an equality of the corresponding chromatic quasisymmetric functions X Γ(λ) (x; q) = X Γ(µ) (x; q) introduced by Shareshian and Wachs [SW16].…”

“…Then vertices v i and v j are joined by an edge if it is possible for entries in cells i and j to form an inversion. This approach was effective in finding combinatorial formulas for unicellular LLT polynomials [Ale21,AP18,HNY20]. Graphs arising from this construction are called unit interval graphs and they have several equivalent characterizations [Gar07].…”