Alexei Borodin, Sergei Korotkikh
We introduce and study an inhomogeneous generalization of the spin $q$-Whittaker polynomials from [Borodin,Wheeler-17]. These are symmetric polynomials, and we prove a branching rule, skew dual and non-dual Cauchy identities, and an integral representation for them. Our main tool is a novel family of deformed Yang-Baxter equations.
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 establish new properties of inhomogeneous spin $q$-Whittaker polynomials, which are symmetric polynomials generalizing $t=0$ Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come not from an $R$-matrix, as is often the case, but from other intertwining operators of $U'_q(\hat{\mathfrak{sl}}_2)$-modules. Using this construction, we are able to prove a Cauchy-type identity for inhomogeneous spin $q$-Whittaker polynomials in full generality. Moreover, we are able to characterize spin $q$-Whittaker polynomials in terms of vanishing at c…
Inhomogeneous $q$-Whittaker Polynomials I: Duality and Expansions
本文提出了一种新的对称多项式家族，统一了多种多项式类型，并通过Yang-Baxter方程推导了转换系数的组合公式。
Inhomogeneous $q$-Whittaker polynomials II: ring theorem and positive specializations
本文研究非均匀$q$-Whittaker多项式，证明其构成特定交换环的基，扩展对称函数环，并探讨正特化与Macdonald正特化的联系，以及由此获得的概率分布。
Orthogonality of spin $q$-Whittaker polynomials
本文证明了非齐次自旋 $q$-Whittaker 多项式关于 $n$ 维环面上 Sklyanin 测度变体的正交性，确立了其作为对称多项式空间的基，并利用非齐次本征关系推广了 Macdonald 多项式的相关性质。
Identity between Restricted Cauchy Sums for the $q$-Whittaker and Skew Schur Polynomials
本研究针对$q$-Whittaker与斜Schur多项式的受限柯西和，借助可积概率中的$q$-Whittaker测度、周期Schur测度及其Fredholm行列式公式，证明了二者间的恒等式，拓展了对称函数柯西恒等式的相关结论。
Spin $q$-Whittaker symmetric polynomials labeled by partitions $λ$ were recently introduced by Borodin and Wheeler (arXiv:1701.06292) in the context of integrable $\mathfrak{sl}_2$ vertex models. They are a one-parameter deformation of the $t=0$ Macdonald polynomials. We present a new, more convenient modification of spin $q$-Whittaker polynomials and find two Macdonald type $q$-difference operators acting diagonally in these polynomials with eigenvalues, respectively, $q^{-λ_1}$ and $q^{λ_N}$ (where $λ$ is the polynomial's label). We study probability measures on interlacing arrays based on s…
Exploiting the fact that the $q$-Whittaker polynomials arise as a specialization of the (modified) Macdonald polynomials, we derive some of their basic properties, and explore interesting identities that they satisfy. We also show how they arise as graded Frobenius characteristics of $§_n$-modules, and give a combinatorial approach to associated Pieri formulas.
Spin $q$-Whittaker polynomials
我们引入并研究了q-Whittaker对称函数的一个单参数推广。这是一个家庭的多元对称多项式，其建设可以看作是一个应用程序的融合过程，从可积晶格模型的顶点模型解释的一个参数推广霍尔-利特尔伍德多项式从[Bor 17，BP 16 a，BP 16 b]。我们证明了分支和Pieri规则，标准和双（斜）柯西求和身份，和一个新的多项式的整表示。
Iterating the skew RSK correspondence discovered by Sagan and Stanley in the late '80s, we define a deterministic dynamics on the space of pairs of skew Young tableaux $(P,Q)$. We find that this skew RSK dynamics displays conservation laws which, in the picture of Viennot's shadow line construction, identify generalizations of Greene invariants. The introduction of a novel realization of $0$-th Kashiwara operators reveals that the skew RSK dynamics possesses symmetries induced by an affine bicrystal structure, which, combined with connectedness properties of Demazure crystals, leads to its lin…
Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial
利用Jack对称多项式给出Virasoro代数的Whittaker向量的显式表示.我们的基本工具是Feigin-Fuchs玻色化和Calogero-Sutherland模型的分裂表达式。
Bijections between different combinatorial models for $q$-Whittaker and modified Hall-Littlewood polynomials
本文为$q$-Whittaker和修正Hall-Littlewood多项式的两种组合公式所诱导的参数集构造保持内容和主指标的双射，并借助charge/cocharge理论证明其正确性。
$q$-Whittaker polynomials: bases, branching and direct limits
该研究围绕$q$-Whittaker多项式展开，分析其三种单项式展开的组合模型，揭示分划分覆盖模式与仿射李代数表示的关联，构建列严格填充模型的投影等概念及模型间双射，并建立与有色格路径形式主义的联系。
In this work we study the relationship between several combinatorial formulas for type $A$ spherical Whittaker functions. These are spherical functions on $p$-adic groups, which arise in the theory of automorphic forms. They depend on a parameter $t$, and are a specialization of Macdonald polynomials, and further specialize to Schur polynomials upon setting $t=0$. There are three types of formulas for these polynomials. The first formula is in terms of so-called alcove walks, works in arbitrary Lie type, and is derived from the Ram-Yip formula for Macdonald polynomials. The second one is in te…
Monomial expansions for $q$-Whittaker and modified Hall-Littlewood polynomials
本文通过构造三种 $q$-Whittaker 多项式模型间的保持权重的双射，应用于局部 Weyl 模极限，得到基本表示的新特征公式，并推广至修正 Hall-Littlewood 情形。
Diagonal operators, $q$-Whittaker functions and rook theory
本文解决了有限域上对角化线性算子作用下具有给定剖面的子空间计数问题，引入了一类由整数分拆对索引的多项式 $b_{\mu\nu}(q)$，并建立了其与半标准杨表、集合划分统计量、$q$-Whittaker 对称函数以及 $q$-车理论的联系。
Lattice Models for Double Whittaker Polynomials and Motivic Chern Classes
本文通过构造依赖两组变量的可解格点模型，统一了 Motivic Chern 类与 Iwahori Whittaker 函数的格点表示，并揭示了 Kazhdan-Lusztig R-多项式形变的递推公式源于 Yang-Baxter 方程。
Macdonald polynomials at t = 0 through twisted multiline queues
该研究通过将多行队列的主指标统计量重新解释为读取词的charge统计量，绕开已有算法得到q-Whittaker多项式的简洁公式，还将结果推广到扭曲多行队列与玻色多行队列，得到相关多项式新公式并恢复经典恒等式。
We prove a `motivic' analogue of the Weyl character formula, computing the Euler characteristic of a line bundle on a generalized flag manifold $G/B$ multiplied either by a motivic Chern class of a Schubert cell, or a Segre analogue of it. The result, given in terms of Demazure-Lusztig (D-L) operators, recovers formulas found by Brubaker, Bump and Licata for the Iwahori-Whittaker functions of the principal series representation of a $p$-adic group. In particular, we obtain a new proof of the classical Casselman-Shalika formula for the spherical Whittaker function. The proofs are based on local…
The inhomogeneous $t$-PushTASEP and Macdonald polynomials
研究有限环上非齐次多物种 $t$-PushTASEP 的平稳概率，证明其与 ASEP 多项式成正比，归一化常数为 Macdonald 多项式，并利用多线图揭示跳跃机制，推导对称性与电流公式。
Inhomogeneous lattice paths, generalized Kostka polynomials and A$_{n-1}$ supernomials
非齐次晶格路径引入有序序列的矩形杨tableaux从而推广最近的工作Kostka多项式由Nakayashiki和山田和Lascoux，Leclerc和Thibon。出于这些工作和Kashiwara理论的晶体基地，我们定义了一个统计路径产生两个新的类多项式。其中之一提供了一个推广的Kostka多项式，而另一个，我们命名为A $_{n-1}$ supernomial，是一个$q$ -变形的产品的舒尔多项式的膨胀系数。许多著名的结果Kostka多项式的扩展导致我们的多项式表示在Littlewood-Richardson表的电荷统计和费米子配置的总和。证明了广义Kostka多项式和A $_{n-1}$超多项式的几个恒等式。最后，将这些超名与Bailey引理联系起来.
The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous $K$-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.
$qt$RSK${}^*$: A probabilistic dual RSK correspondence for Macdonald polynomials
本文引入了依赖参数$q$和$t$的概率对偶RSK对应$qt$RSK${}^*$，基于Fomin增长图首次给出了Macdonald多项式对偶Cauchy恒等式的tableaux理论证明，并证明了在Jack多项式限制下插入tableaux的字母交换不变性。
Consider a lattice of n sites arranged around a ring, with the $n$ sites occupied by particles of weights $\{1,2,\dots,n\}$; the possible arrangements of particles in sites thus corresponds to the $n!$ permutations in $S_n$. The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on the set of permutations, in which two adjacent particles of weights $i<j$ swap places at rate $x_i - y_{n+1-j}$ if the particle of weight $j$ is to the right of the particle of weight $i$. (Otherwise nothing happens.) In the case that $y_i=0$ for all $i$, the stationary distributi…
Consider a lattice of n sites arranged around a ring, with the $n$ sites occupied by particles of weights $\{1,2,\dots,n\}$; the possible arrangements of particles in sites thus corresponds to the $n!$ permutations in $S_n$. The \emph{inhomogeneous totally asymmetric simple exclusion process} (or TASEP) is a Markov chain on the set of permutations, in which two adjacent particles of weights $i<j$ swap places at rate $x_i - y_{n+1-j}$ if the particle of weight $j$ is to the right of the particle of weight $i$. (Otherwise nothing happens.) In the case that $y_i=0$ for all $i$, the stationary dis…
A new generalisation of Macdonald polynomials
本文引入了一类新的对称多元多项式族，其系数是两个参数$(q,t)$的亚纯函数和另一个参数$(u,v)$的多项式.我们评估这些多项式明确作为一个矩阵产品。在$u=v=0$，他们减少到麦克唐纳多项式，而在$q=0$，$u=v=s$，他们恢复家庭的非齐次对称函数最初介绍了鲍罗丁。
Two Littlewood identities for fully inhomogeneous spin Hall-Littlewood symmetric rational functions
本文研究了完全非均匀自旋Hall-Littlewood对称有理函数的两个新恒等式，通过路径配置的分区函数定义，建立了Hall-Littlewood对称多项式的新恒等式，并探讨了其与修改后的Robbins多项式之间的联系。
We prove a positivity result for interpolation polynomials that was conjectured by Knop and Sahi. These polynomials were first introduced by Sahi in the context of the Capelli eigenvalue problem for Jordan algebras, and were later shown to be related to Jack polynomials by Knop-Sahi and Okounkov-Olshanski. The positivity result proved here is an inhomogeneous generalization of Macdonald's positivity conjecture for Jack polynomials. We also formulate and prove the non-symmetric version of the Knop-Sahi conjecture, and in fact we deduce everything from an even stronger positivity result. This la…
Maximal Cliques in Scale-Free Random Graphs
该研究分析了三种随机图模型中极大团的数量，给出不同模型下的上下界，通过实验解决了几何非均匀随机图极大团数量的经验矛盾，揭示其超多项式下界及极端规模下的渐近行为。
Sahi, Stokman, and Venkateswaran have constructed, for each positive integer $n$, a family of Laurent polynomials depending on parameters $q$ and $k$ (in addition to $\lfloor n/2\rfloor$ "metaplectic parameters"), such that the $n=1$ case recovers the nonsymmetric Macdonald polynomials and the $q\rightarrow\infty$ limit yields metaplectic Iwahori-Whittaker functions with arbitrary Gauss sum parameters. In this paper, we study these new polynomials, which we call SSV polynomials, in the case of $GL_r$. We apply a result of Ram and Yip in order to give a combinatorial formula for the SSV polynom…
Nonsymmetric interpolation Macdonald polynomials and g_n basic hypergeometric series
Knop-Sahi插值Macdonald多项式是著名的Macdonald多项式的非齐次和非对称推广。本文应用插值Macdonald多项式研究了一类新的g_n型基本超几何级数。我们的主要结果包括一个新的q-二项式定理，新的q-高斯和，以及g_n级数的几个变换公式。
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。