Maciej Kowalski
We give a combinatorial formula for LLT cumulants of unicellular unilevelled shapes in terms of parking functions and Cayley trees. Our formula implies previously conjectured Schur positivity.
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It is known that unicellular LLT polynomials are related to the quasi-symmetric chromatic polynomials of certain graphs by the $(t-1)$-transform of symmetric functions. We investigate the extension of this transformation to various combinatorial Hopf algebras and prove a noncommutative version of this property.
In this work, we generalize and utilize the linear relations of LLT polynomials introduced by Lee \cite{Lee}. By using the fact that the chromatic quasisymmetric functions and the unicellular LLT polynomials are related via plethystic substitution and thus they satisfy the same linear relations, we can apply the linear relations to both sets of functions. As a result, in the chromatic quasisymmetric function side, we find a class of $e$-positive graphs, called \emph{melting lollipop graphs}, and explicitly prove the $e$-unimodality. In the unicellular LLT side, we obtain Schur expansion formul…
Expanding the unicellular LLT polynomials of two-headed melting lollipops into ribbon Schurs
本文针对双头融化棒棒糖图，证明其单细胞 LLT 多项式可展开为带状 Schur 函数的简单公式，推广了 Huh、Nam 和 Yoo 对融化棒棒糖图的 Schur 展开结果。
LLT polynomials are $q$-analogues of product of Schur functions that are known to be Schur-positive by Grojnowski and Haiman. However, there is no known combinatorial formula for the coefficients in the Schur expansion. Finding such a formula also provides Schur positivity of Macdonald polynomials. On the other hand, Haiman and Hugland conjectured that LLT polynomials for skew partitions lying on $k$ adjacent diagonals are $k$-Schur positive, which is much stronger than Schur positivity. In this paper, we prove the conjecture for $k=2$ by analyzing unicellular LLT polynomials. We first present…
The solution of Shareshian-Wachs conjecture by Brosnan-Chow linked together the cohomology of regular semisimple Hessenberg varieties and graded chromatic symmetric functions on unit interval graphs. On the other hand, it is known that unicellular LLT polynomials have similar properties to graded chromatic symmetric functions. In this paper, we link together the unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties introduced by Ayzenberg-Buchstaber. We prove their palindromicity from topological viewpoint. We also show that modules of a symmetric group generated by f…
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 and 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 an…
Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials
本文研究正则半单Hessenberg簇双流形的几何，证明二者通过显式爆破与纤维丛映射关联，取上同调后直接证明模律，确立LLT型Shareshian-Wachs猜想。
LLT Polynomials and Hecke Algebra Traces
该研究揭示单细胞LLT多项式系数与Hecke代数迹的关联，给出其用传统迹基、归纳法及Kazhdan-Lusztig R-多项式的表达形式，推进相关代数组合研究。
A combinatorial formula for LLT cumulants of melting lollipops in terms of spanning trees
本文给出熔化棒棒糖图LLT累积量的生成树正组合公式，肯定其正性并独立证明Schur正性，完全图情形下用停车函数表达。
We prove that the combinatorial side of the "Rational Shuffle Conjecture" provides a Schur-positive symmetric polynomial. Furthermore, we prove that the contribution of a given rational Dyck path can be computed as a certain skew LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel and Ulyanov. The corresponding skew diagram is described explicitly in terms of a certain (m,n)-core.
We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as vertical strip --- in particular, unicellular LLT polynomials. We show that there are parallel phenomena regarding $e$-positivity of these two families of polynomials. In particular, we give several examples where the LLT polynomials behave like a "mirror image" of the chromatic quasisymmetric counterpart. The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials. This circular extensions of LLT polynomials…
For an indifference graph $G$ we define a symmetric function of increasing spanning forests of $G$. We prove that this symmetric function satisfies certain linear relations, which are also satisfied by the chromatic quasisymmetric function and unicellular LLT polynomials. As a consequence we give a combinatorial interpretation of the coefficients of the LLT polynomial in the elementary basis (up to a factor of a power of $(q-1)$), strengthening the description given by Alexandersson and Sulzgruber.
A parking function interpretation for $(-1)^{k}\nabla m_{2^{k}1^{l}}$
本文通过建立递归证明 $(-1)^{k}\nabla m_{2^{k}1^{l}}$ 的 Schur 正性，并利用组合 shuffle 定理给出其停车函数解释。
The modified Macdonald polynomials introduced by Garsia and Haiman (1996) have many remarkable combinatorial properties. One such class of properties involves applying the $\nabla$ 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 $\nabla e_n$ 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…
$P$-partitions and $p$-positivity
利用$\alpha$-单峰集的组合学，本文建立了拟对称函数理论中基本基展开与反向$P$-划分生成函数正展开两个新结果，并应用于多种多项式与函数，改进文献中的正性结论。
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