Sylvie Corteel, Andrew Gitlin, David Keating, Jeremy Meza
We describe a novel Yang-Baxter integrable vertex model. From this vertex model we construct a certain class of partition functions that we show are equal to the LLT polynomials of Lascoux, Leclerc, and Thibon. Using the vertex model formalism, we give alternate proofs of many properties of these polynomials, including symmetry and a Cauchy identity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We describe a Yang-Baxter integrable vertex model, which can be realized as a degeneration of a vertex model introduced by Aggarwal, Borodin, and Wheeler. From this vertex model, we construct a certain class of partition functions that we show are essentially equal to the super ribbon functions of Lam. Using the vertex model formalism, we give proofs of many properties of these polynomials, namely a Cauchy identity and generalizations of known identities for supersymmetric Schur polynomials.
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…
We prove that two horizontal-strip LLT polynomials are equal if the associated weighted graphs defined by the author in a previous paper are isomorphic. This provides a sufficient condition for equality of horizontal-strip LLT polynomials and yields a well-defined LLT polynomial indexed by a weighted graph. We use this to prove some new relations between LLT polynomials and we explore a connection with extended chromatic symmetric functions.
We introduce a solvable lattice model for supersymmetric LLT polynomials, also known as super LLT polynomials, based upon particle interactions in super n-ribbon tableaux. Using operators on a Fock space, we prove a Cauchy identity for super LLT polynomials, simultaneously generalizing the Cauchy and dual Cauchy identities for LLT polynomials. Lastly, we construct a solvable semi-infinite Cauchy lattice model with a surprising Yang-Baxter equation and examine its connections to the Cauchy identity.
In this text we introduce and analyze families of symmetric functions arising as partition functions for colored fermionic vertex models associated with the quantized affine Lie superalgebra $U_q \big( \widehat{\mathfrak{sl}} (1 | n) \big)$. We establish various combinatorial results for these vertex models and symmetric functions, which include the following. (1) We apply the fusion procedure to the fundamental $R$-matrix for $U_q \big( \widehat{\mathfrak{sl}} (1 | n) \big)$ to obtain an explicit family of vertex weights satisfying the Yang-Baxter equation. (2) We define families of symmetric…
Coloured corner processes from asymptotics of LLT polynomials
研究秩 $n \geq 1$ 的 LLT 对称多项式 Cauchy 恒等式导出的概率测度，利用费米子格点模型和围道积分表示，证明其在 $n>1$ 时渐近分裂为 $n$ 个 GUE 角过程乘积的连续部分与含交错着色的离散部分。
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 recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial $G_λ(x;q)$ in some special cases. We associate a weighted graph $Π$ to $λ$ and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of $G_λ(x;q)$ whenever $Π$ is triangle-free. We also prove that the largest power of $q$ in the LLT polynomial is the total edge weight of our graph.
Lascoux, Leclerc and Thibon\cite{LLT} introduced a family of symmetric polynomials, called LLT polynomials. We prove a $q$-multinomial expansion of the coefficients of LLT polynomials in the case where $ \boldsymbolμ = \underbrace{(μ,...,μ)}_{n}$ and define a $q$-analog of a sum of the plethysm multiplicities.
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…
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…
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…
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。
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…
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。
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 identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies $Λ(X^{m,n})\subset \mathcal{E}$ of the algebra of symmetric functions embedded in the elliptic Hall algebra $\mathcal{E}$ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the $\nabla$ operator applied to any LLT polynomial. In particular, we obtain a formula for $\nabla ^m s_λ$ which serves as a starting point for our proof of the Loehr-Warrington conjecture in a companion…
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。
In this article we prove the $e$-positivity of $G_{\mathbfν}[X;q+1]$ when $G_{\mathbfν}[X;q]$ is a vertical strip LLT polynomial. This property has been conjectured by Alexandersson and Panova, and by Garsia, Haglund, Qiu and Romero, and it implies several $e$-positivities conjectured by them and also by Bergeron. We make use of a result of Carlsson and Mellit that shows that a vertical strip LLT polynomial can be obtained by applying certain compositions of operators of the Dyck path algebra to the constant $1$. Our proof gives in fact an algorithm to expand these symmetric functions in the e…
A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
该研究给出竖直带LLT多项式的新组合刻画，证明其按初等对称函数的显式组合展开式，验证相关猜想并重现已有显正性结果，还得到单位区间图色拟对称函数的新刻画。
Flagged LLT polynomials, nonsymmetric plethysm, and nonsymmetric Macdonald polynomials
本文引入非对称标记 LLT 多项式与非对称 plethysm 算子，重构非对称 Macdonald 多项式公式，并提出更强的非对称 Macdonald 正性猜想。
Equivalences of LLT polynomials via lattice paths
该研究通过构造顶点模型构型的双射，证明了两组边界条件互换的LLT多项式在相差整体t因子意义下相等，还可系统确定LLT多项式族的线性关系。
The nabla operator, Hall--Littlewood polynomials and LLT polynomials
本文证明了关于nabla算子、Hall–Littlewood多项式与LLT多项式的两个猜想，补充了Carlsson和Mellit已证的结论，还将结果推广到Schiffmann代数框架。
We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam's algebra of ribbon Schur operators. Combining this result with the expression of Haglund, Haiman, and Loehr for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Macdonald polynomials indexed by a shape with 3 columns.
In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's fundamental quasisymmetric functions so that the sum of the weights of a connected component is a single Schur function. In this paper, we improve on Assaf's axiomatization of such graphs, giving locally testable criteria that are more easily verified by computers. We further advance the theory of dual equivalence graphs by describing a broader class of graphs that correspond…
We define a D_0 graph to be a graph whose vertex set is a subset of permutations of n, with edges of the form ...bac... <--> ...bca... or ...acb... <--> ...cab... (Knuth transformations), or ...bac... <--> ...acb... or ...bca... <--> ...cab... (rotation transformations), such that whenever the Knuth and rotation transformations at positions i-1, i, i+1 are available at a vertex, exactly one of these is an edge. The generating function of such a graph is the sum of the quasisymmetric functions associated to the descent sets of its vertices. Assaf studied D_0 graphs in the paper *Dual equivalenc…
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.
Inspired by the $k$-inversion statistic for LLT polynomials, we define a $k$-inversion number and $k$-descent set for words. Using these, we define a new statistic on words, called the $k$-major index, that interpolates between the major index and inversion number. We give a bijective proof that the $k$-major index is equidistributed with the major index, generalizing a classical result of Foata and rediscovering a result of Kadell. Inspired by recent work of Haglund and Stevens, we give a partial extension of these definitions and constructions to standard Young tableaux. Finally, we give an …
We provide new approaches to prove identities for the modified Macdonald polynomials via their LLT expansions. As an application, we prove a conjecture of Haglund concerning the multi-$t$-Macdonald polynomials of two rows.
The nonsymmetric compositional Delta theorem
本文基于Blasiak等人对标志性LLT多项式理论，推导出非对称身份，并引入非对称∇和τ*算子，提出新的算子形式，通过Weyl对称化恢复原始组合Delta定理，并提出关于稳定原子正性猜想。
A raising operator formula for Macdonald polynomials
本文利用LLT多项式上的 $\nabla$ 公式和Haglund-Haiman-Loehr公式，得到修正Macdonald多项式的显式升算子公式，并推广提出 $1,n$-Macdonald多项式及其Schur系数正性猜想。
We describe a general algorithm for generating various families of ribbon tableaux and computing their spin polynomials. This algorithm is derived from a new matricial coding. An advantage of this new notation lies in the fact that it permits one to generate ribbon tableaux with skew shapes. This algorithm permits us to compute quickly big LLT polynomials in MuPAD-Combinat.
Colored vertex models and $k$-tilings of the Aztec diamond
本文利用彩色顶点模型计算 Aztec 钻石 $k$-铺砌的生成多项式，证明无相互作用铺砌与普通铺砌的双射，并求得相关北极曲线。
We study solvable lattice models associated to canonical Grothendieck polynomials and their duals. We derive inversion relations and Cauchy identities.
The censored stochastic six-vertex model and parabolic Kazhdan--Lusztig $R$-polynomials
引入删失随机六顶点模型，证明参数$b_1<b_2$时从初始条件${1}_{x>0}$出发的模型被阻塞测度随机占优，利用抛物型Kazhdan-Lusztig $R$-多项式证明，并发现交织关系。
The determinantal form of the partition function of the 6-vertex model with domain wall boundary conditions was given by Izergin. It is known that for a special value of the crossing parameter the partition function reduces to a Schur polynomial. Caradoc, Foda and Kitanine computed the partition function of the higher spin generalization of the 6-vertex model. In the present work it is shown that for a special value of the crossing parameter, referred to as the combinatorial point, the partition function reduces to a Macdonald polynomial.
We construct an integrable colored five-vertex model whose partition function is a Lascoux atom based on the five-vertex model of Motegi and Sakai [arXiv:1305.3030] and the colored five-vertex model of Brubaker, the first author, Bump, and Gustafsson [arXiv:1902.01795]. We then modify this model in two different ways to construct a Lascoux polynomial, yielding the first known combinatorial interpretation of a Lascoux polynomial and atom. Using this, we prove a conjectured combinatorial interpretation in terms of set-valued tableaux of a Lascoux polynomial and atom due to Pechenik and the secon…
We introduce and study several combinatorial properties of a class of symmetric polynomials from the point of view of integrable vertex models in finite lattice. We introduce the $L$-operator related with the $U_q(sl_2)$ $R$-matrix, and construct the wavefunctions and their duals. We prove the exact correspondence between the wavefunctions and symmetric polynomials which is a quantum group deformation of the Grothendieck polynomials. This is proved by combining the matrix product method and an analysis on the domain wall boundary partition functions. As applications of the correspondence betwe…
We study certain symmetric polynomials, which as very special cases include polynomials related to the supersymmetric eight-vertex model, and other elliptic lattice models with $Δ=\pm 1/2$. In this paper, which is the first part of a series, we study the behaviour of the polynomials at special parameter values, which can be identified with cusps of the modular group $Γ_0(12)$. In subsequent papers, we will show that the polynomials satisfy a non-stationary Schrödinger equation related to the Knizhnik--Zamolodchikov--Bernard equation and that they give a four-dimensional lattice of tau function…
Starting from an integrable rank-$n$ vertex model, we construct an explicit family of partition functions indexed by compositions $μ= (μ_1,\dots,μ_n)$. Using the Yang-Baxter algebra of the model and a certain rotation operation that acts on our partition functions, we show that they are eigenfunctions of the Cherednik-Dunkl operators $Y_i$ for all $1 \leq i \leq n$, and are thus equal to nonsymmetric Macdonald polynomials $E_μ$. Our partition functions have the combinatorial interpretation of ensembles of coloured lattice paths which traverse a cylinder. Applying a simple bijection to such pat…
We study concentration properties of vertex degrees of $n$-dimensional Erdos-Rényi random graphs with the edge probability $ρ/n$ by means of high moments of these random variables in the limit when $n$ and $ρ$ tend to infinity. These moments are asymptotically close to one-variable Bell polynomials ${\cal B}_k(ρ), k\in {\bf N}$ that represent moments of the Poisson probability distribution ${\cal P}(ρ)$. We study asymptotic behavior of the Bell polynomials and modified Bell polynomials for large values of $k$ and $ρ$ with the help of the local limit theorem for auxiliary random variables. Usin…
We study Hall-Littlewood polynomials using an integrable lattice model of $t$-deformed bosons. Working with row-to-row transfer matrices, we review the construction of Hall-Littlewood polynomials (of the $A_n$ root system) within the framework of this model. Introducing appropriate double-row transfer matrices, we extend this formalism to Hall-Littlewood polynomials based on the $BC_n$ root system, and obtain a new combinatorial formula for them. We then apply our methods to prove a series of refined Cauchy and Littlewood identities involving Hall-Littlewood polynomials. The last two of these …
A generalized vertex join of a graph is obtained by joining an arbitrary multiset of its vertices to a new vertex. We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees, and by duality we find the flow polynomials of arbitrary outerplanar graphs. We also present closed formulas for the chromatic and flow polynomials of vertex joins of cliques and cycles, otherwise known as "generalized wheel" graphs.
The paper presents a polynomial time approximation schema for the edge-weighted version of maximum k-vertex cover problem in bipartite graphs.
New relations for the vertex polynomial
该研究将顶点多项式扩展到任意度的图，证明了图含二边形等时的局部关系，丰富了顶点多项式的相关理论。
The classical Dodgson identity can be interpreted as a quadratic identity of spanning forest polynomials, where the spanning forests used in each polynomial are defined by how three marked vertices are divided among the component trees. We prove an analogous result with four marked vertices.
Introducing a vertex polynomial invariant for embedded graphs
针对嵌入图扭曲对偶中的顶点计数问题，提出顶点多项式这一轨道分布生成函数，建立其边界分量等价形式、删边/收缩递归及花束图的带符号相交图依赖，并给出与interlace多项式和拓扑转移多项式相联系的拓扑解释。
For a hypergraph $\mathcal H$, we consider the edge-induced and vertex-induced subhypergraph polynomials and study their relation. We use this relation to prove that both polynomials are reconstructible, and to prove a theorem relating the Hilbert series of the Stanley-Reisner ring of the independent complex of $\mathcal H$ and the edge-induced subhypergraph polynomial. We also consider reconstruction of some algebraic invariants of $\mathcal H$.
We present Pieri rules for the Jack polynomials in superspace. The coefficients in the Pieri rules are, except for an extra determinant, products of quotients of linear factors in $α$ (expressed, as in the usual Jack polynomial case, in terms of certain hook-lengths in a Ferrers' diagram). We show that, surprisingly, the extra determinant is related to the partition function of the 6-vertex model. We give, as a conjecture, the Pieri rules for the Macdonald polynomials in superspace.
For a undirected simple graph $G$, let $d_i(G)$ be the number of $i$-element dominating vertex set of $G$. The domination polynomial of the graph $G$ is defined as $$D(G, x) = \sum_{i = 1}^n d_i(G)x^i.$$ Alikhani and Peng conjectured that $D(G, x)$ is unimodal for any graph $G$. Answering a proposal of Beaton and Brown, we show that $D(G, x)$ is unimodal when $G$ has at least $2^{13}$ vertices and has a universal vertex, which is a vertex adjacent to any other vertex of $G$. We further determine possible locations of the mode.
Chromatic polynomials of signed graphs and dominating-vertex deletion formulae
本文通过构造共享底层图和色多项式的非切换同构符号图回答了 Zaslavsky 的问题，推导了符号完全图连接的色多项式闭式表达，并引入双变量色多项式及其支配顶点删除递归公式，证明其在特定符号阈值图族中可刻画同构。
The computational complexity of the Vertex Coloring problem is known for all hereditary classes of graphs defined by forbidding two connected five-vertex induced subgraphs, except for seven cases. We prove the polynomial-time solvability of four of these problems: for ($P_5$, dart)-free graphs, ($P_5$, banner)-free graphs, ($P_5$, bull)-free graphs, and (fork, bull)-free graphs.
We prove a quantitative, finitary version of Trofimov's result that a connected, locally finite vertex-transitive graph G of polynomial growth admits a quotient with finite fibres on which the action of Aut(G) is virtually nilpotent with finite vertex stabilisers. We also present some applications. We show that a finite, connected vertex-transitive graph G of large diameter admits a quotient with fibres of small diameter on which the action of Aut(G) is virtually abelian with vertex stabilisers of bounded size. We also show that G has moderate growth in the sense of Diaconis and Saloff-Coste, …
A family of vertex operators that generalizes those given by Jing for the Hall-Littlewood symmetric functions is presented. These operators produce symmetric functions related to the Poincare polynomials referred to as generalized Kostka polynomials in the same way that Jing's operator produces symmetric functions related to Kostka-Foulkes polynomials. These operators are then used to derive commutation relations and new relations involving the generalized Kostka coefficients. Such relations may be interpreted as identities in the (GL(n) x C^*)-equivariant K-theory of the nullcone.
We give several new formulas which are useful for Schubert Calculus associated with the orthogonal groups and related orthogonal degeneracy loci.
We study the effect of linear transformations on quantum fields with applications to vertex operator presentations of symmetric functions. Properties of linearly transformed quantum fields and corresponding transformations of Hall-Littlewood polynomials are described, including preservation of commutation relations, stability, explicit combinatorial formulas and generating functions. We prove that specializations of linearly transformed Hall-Littlewood polynomials describe all polynomial tau functions of the KP and the BKP hierarchy. Examples of linear transformations are related to multiparam…
A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a $P_5$-free graph with clique number $ω\ge 3$ has chromatic number at most $ω^{\log_2(ω)}$. The best previous result was an exponential upper bound $(5/27)3^ω$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdos-Hajnal conjecture holds for $P_5$, which is the smallest open case. Thus there is great interest in whether there is a polynomial bound for $P_5$-free graphs, and our result is an attempt to approach that.
The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this Note we formulate a q,t-deformation of this n-point function. The key operator used in our formulation arises from the theory of Macdonald polynomials and affords a vertex operator interpretation. We obtain closed formulas for the n-point functions when n =1,2 in terms of the basic hypergeometric functions. We further generalize the q,t-deformed n-point function to more general vertex operators.
Vertex models for the product of a permuted-basement Demazure atom and a Schur polynomial
本文提出了展开置换基Demazure原子与Schur多项式乘积的首个正组合规则，通过推导着色顶点模型并扩展Zinn-Justin的技术，在单一框架下给出了包含已知特例的新公式，并利用特定边界条件下的Yang-Baxter方程完成证明。
Spectral moments and characteristic polynomials of vertex expansion hypergraphs of graphs
本文通过$s$-倍闭游走刻画图的顶点扩张超图的谱矩，并由此推导其特征多项式。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。