Noncommutative Bessel symmetric functions
Jean-Christophe Novelli, Jean-Yves Thibon
考虑0-Hecke代数模的张量积，得到了非交换对称函数代数中Bessel J-函数的自然类似物。这提供了一个简单的解释各种组合性质的贝塞尔函数。
The consideration of tensor products of 0-Hecke algebra modules leads to natural analogs of the Bessel J-functions in the algebra of noncommutative symmetric functions. This provides a simple explanation of various combinatorial properties of Bessel functions.
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We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then projecting the coefficients to $QSym$. By duality, we obtain a basis of noncommutative symmetric functions, for which a product formula and a recurrence in the form of a combinatorial complex are obtained. This basis allows to identify noncommutative symmetric functions with the quotient of FQSym induced by the pattern-replacement relation $321 \equiv 231$ and $312 …
Noncommutative symmetric functions and quasi-symmetric functions with two and more paramters
我们定义了非交换对称函数和拟对称函数的双参数族，它们似乎是Macdonald对称函数在这些设置中的适当类似物。
In the Hopf algebra of symmetric functions, Sym, the basis of Schur functions is distinguished since every Schur function is isomorphic to an irreducible character of a symmetric group under the Frobenius characteristic map. In this note we show that in the Hopf algebra of noncommutative symmetric functions, Nsym, of which Sym is a quotient, the recently discovered basis of noncommutative Schur functions exhibits that every noncommutative Schur function is isomorphic to a noncommutative irreducible character of a symmetric group when working in noncommutative character theory. We simultaneousl…
We investigate the connections between various noncommutative analogues of Hall-Littlewood and Macdonald polynomials, and define some new families of noncommutative symmetric functions depending on two sequences of parameters.
Noncommutative Symmetric Functions Associated with a Code, Lazard Elimination, and Witt Vectors
Witt向量的泛环的构造与自由幺半群的Lazard因子分解通过一个非对易的类似物有关。这是通过将非交换对称函数的特殊化与代码相关联来完成的。
We present a simple way to derive the results of Diaconis and Fulman [arXiv:1102.5159] in terms of noncommutative symmetric functions.
We define new families of noncommutative symmetric functions and quasi-symmetric functions depending on two matrices of parameters, and more generally on parameters associated with paths in a binary tree. Appropriate specializations of both matrices then give back the two-vector families of Hivert, Lascoux, and Thibon and the noncommutative Macdonald functions of Bergeron and Zabrocki.
Noncommutative Symmetric Functions and Lagrange Inversion
计算了0-Hecke代数自然作用在停车函数上的非交换Frobenius特征，并作为推论得到了非交换Lagrange反演公式的各种形式.
We introduce a new pair of mutually dual bases of noncommutative symmetric functions and quasi-symmetric functions, and use it to derive generalizations of several results on the reduced incidence algebra of the lattice of noncrossing partitions. As a consequence, we obtain a quasi-symmetric version of the Farahat-Higman algebra.
Noncommutative Symmetric Functions VII: Free Quasi-Symmetric Functions Revisited
证明了自由拟对称函数的一个柯西恒等式，并将其应用于各种基的研究。本文还讨论了自由Weyl公式和分裂公式的推广。
The chromatic noncommutative symmetric function of oriented trees
研究有向树的色非交换对称函数能否区分非同构的有向树，并证明其可重构有向星、有向双星、某些有向毛虫树和某些有向路径。
Noncommutative Analogs of Monomial Symmetric Functions, Cauchy Identity, and Hall Scalar Product
本文介绍了单项对称函数的非交换类似物和基本非交换对称函数。带状Schur函数在这两种基上的展开都是非负的。有了这些函数，我们可以导出一个非交换柯西恒等式，也可以研究柯西恒等式所隐含的非交换标量积。这个标积似乎是交换理论中Hall标积的非对易类比。
Noncommutative symmetric functions VI: Free quasi-symmetric functions and related algebras
本文主要研究与对称函数有关的几种代数，它们的线性基可以被各种组合对象标记：置换（自由拟对称函数）、标准Young tableaux（自由对称函数）和压缩整数矩阵（矩阵拟对称函数）。自由拟对称函数为0-Hecke代数上的某类模提供了一种非交换Frobenius特征。讨论了不可分解的$H_n(0)$ -模的新例子，并对小的$n$计算了$H_n(0)$的同调性质.最后，矩阵拟对称函数的代数被解释为卷积代数。
On the S_n-module structure of the noncommutative harmonics
使用一个非交换模拟Chevalley的分解多项式到对称多项式倍coinvariants由于Bergeron，Littenauer，Rosas，和Zabrocki我们计算的分级Frobenius系列的两套非交换谐波相对于左行动的对称群（作用于变量）。我们利用这些结果导出n元自由李代数的包络代数的Frobenius级数。
Noncommutative symmetric functions and skewing operators
本文发展非交换对称函数理论以研究偏斜算子，利用该机制复现Littlewood–Richardson规则，并为色拟对称函数建立推广Harada–Precup递推的递推关系。
Permutation statistics related to a class of noncommutative symmetric functions and generalizations of the Genocchi numbers
我们证明了第三作者[L. Tevlin，Proc. FPSAC'07，Tianjin]在非交换对称函数的两种新基上：带状基的转移阵具有非负整系数.这是通过两个组合值的统计排列和包装的话，推广的组合Genocchi数。
Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions
本文研究了对称函数的霍普夫代数的两个推广。非交换对称函数的霍普夫代数及其对偶，拟对称函数的霍普夫代数。重点是令人难以置信的丰富结构的霍普夫代数的对称函数和问题的结构和性质有很好的类似物的非交换对称函数和/或准对称函数。本文试图调查正在进行的调查，在这一主题所决定的知识和兴趣的作者。讨论了许多悬而未决的问题。
Grothendieck bialgebras, Partition lattices and symmetric functions in noncommutative variables
证明了分划格代数的半塔的Grothendieck双代数与非交换变量对称函数双代数的分次对偶同构。特别是这种同构挑出一个典型的新的基础上的对称函数的非交换变量，这将是一个类似的舒尔函数的基础上，这双代数。
Record compositions of alternating permutations and noncommutative symmetric functions
解决阿姆德贝汉等人关于记录组合是否有类似理论的开放问题，证明具有特定记录组合的\(\{1,\dots,2n\}\)交错排列数公式，该数是芽对称函数提升到非交换对称函数展开系数，且对特定芽序列有类似细化，结果由AxiomProver在Lean中验证。
Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions
我们引入自由拟对称函数的霍普夫代数的类似物，其基由色置换标记。当色集是半群时，可以引入内积。这导致了与圈积$Γ\wr\SG_n$相关的广义下降代数的构造和相应的拟对称函数的推广。相关的霍普夫代数作为麦克马洪的多对称函数的自然类似物出现。由此得到了一般多重对称函数的内积。我们将这些构造推广到了色parking函数、色非交叉剖分和B型parking函数的霍普夫代数。
A Noncommutative Symmetric System over the Grossman-Larson Hopf Algebra of Labeled Rooted Trees
本文在标号有根树的Grossman-Larson霍普夫代数上构造了一个非交换对称（${\mathcal N}$ CS）系统.利用由某些非交换对称函数的母函数构成的${\mathcal N}$ CS系统的泛性质，利用标号有根树得到了非交换对称函数的一个特殊化.在分次同态的基础上，得到了标号根森林的Connes-Kreimer霍普夫代数到拟对称函数的霍普夫代数的分次霍普夫代数同态.给出并证明了所构造的${\mathcal N}$ CS系统的第三生成函数的系数与有根树的阶多项式的关系。
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.
A Combinatorial Perspective on the Noncommutative Symmetric Functions
该研究从组合视角出发，给出非交换对称函数$\textbf{NSym}$各基的具体非交换变量函数表示（无需拟行列式），并探究传统对称函数基变换的两种阐释的非交换类似形式。
Let $\CRF_S$ denote the category of $S$-colored rooted forests, and $\H_{\CRF_S}$ denote its Ringel-Hall algebra as introduced in \cite{KS}. We construct a homomorphism from a $K^+_0 (\CRF_S)$--graded version of the Hopf algebra of noncommutative symmetric functions to $\H_{\CRF_S}$. Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a $K^+_0 (\CRF_S)$--graded version of the algebra of quasisymmetric functions. This homomorphism is a refinement of one considered by W. Zhao in \cite{Z}.
The primitives of the Hopf algebra of noncommutative symmetric functions
设NSymm是整数上非交换对称函数的霍普夫代数.本文用递推公式描述了它的整数基元李代数Prim（NSymm）。对于素数基（NSymm）的每个基元，都有一个递归给定的整除幂级数，这给出了拟对称函数代数在整数上是自由的定理的另一个证明.
On the Hopf algebra of noncommutative symmetric functions in superspace
本文研究了超空间中非交换对称函数的Hopf代数sNSym，通过引入本原元素、推广初等与幂和函数、构造非交换Ribbon Schur函数及其对偶基本拟对称函数，并证明sNSym可实现为树的Hopf代数。
Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can be lifted at the level of noncommutative symmetric functions or free quasi-symmetric functions. The results take the form of algebraic identities for type B noncommutative symmetric functions, noncommuta…
The noncommutative geode
本文从非交换对称函数的角度研究测地线及其推广。
A Noncommutative Chromatic Symmetric Function
Stanley将一个图G与一个对称函数X_G联系起来，X_G在变量的某种特殊化下可约化为G的色多项式。然后，他证明了各种定理推广的结果约色多项式，以及新的，不能解释在这一水平。不幸的是，X_G不满足删除-压缩律，这使得它很难应用归纳法。我们在非交换变量中引入了一个对称函数，当变量允许交换时，它确实有这样一个规律，并且专门针对X_G。这使我们能够进一步推广一些斯坦利的定理，并证明他们在一个统一和直接的方式。此外，我们还对Stanley和Stembridge的（3+1）-free猜想作了一些改进.
We define an action of the 0-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their (q,t)-analogues introduced by Bergeron and Zabrocki, and to a more general family of noncommutative symmetric functions having parameters associated with paths in binary trees introduced recently by Lascoux, Novelli, and Thibon. We also obtain multivariate quasisymmetric function identities, which specialize to re…
Chromatic quasisymmetric functions and noncommutative $P$-symmetric functions
本文针对自然单位区间序，利用堆和局部翻转操作研究色拟对称函数，建立其与非交换对称函数的对偶，并得到正展开及e-正性猜想的部分结果。
Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation
本文将Shareshian-Wachs色准对称函数提升至非交换变量代数，提出兼容Haglund-Wilson公式的非交换Macdonald多项式及多重$t$版本，并猜想矩形分拆下的多重$t$ Macdonald多项式可由Hecke代数中Yang-Baxter元素的等变迹获得。
Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables
我们在Rosas和Sagan最近作为向量空间研究的非交换对称函数空间上引入了一个自然的霍普夫代数结构。这个代数的基由集合划分索引。我们发现，存在一个自然包含的霍普夫代数的非交换对称函数索引的组合在这个更大的空间。我们也认为这个代数作为一个子空间的非交换多项式，并用它来了解结构的空间的谐波和协不变关于这个集合的非交换多项式。
The machinery of noncommutative Schur functions is a general approach to Schur positivity of symmetric functions initiated by Fomin-Greene. Hwang recently adapted this theory to posets to give a new approach to the Stanley-Stembridge conjecture. We further develop this theory to prove that the symmetric function associated to any $P$-Knuth equivalence graph is Schur positive. This settles a conjecture of Kim and the third author, and refines results of Gasharov, Shareshian-Wachs, and Hwang on the Schur positivity of chromatic symmetric functions.
Noncommutative Shifted Symmetric Functions
该研究引入基于整数索引移位参数序列的非交换移位对称函数环，构造其基与多类经典公式的类似物，给出非交换变量有理函数实现，丰富了非交换对称函数理论。
We define a new family of noncommutative Bell polynomials in the algebra of free quasi-symmetric functions and relate it to the dual immaculate basis of quasi-symmetric functions. We obtain noncommutative versions of Grinberg's results [Canad. J. Math. 69 (2017), 21--53], and interpret them in terms of the tridendriform structure of WQSym. We then present a variant of Rey's self-dual Hopf algebra of set partitions [FPSAC'07, Tianjin] adapted to our noncommutative Bell polynomials and give a complete description of the Bell equivalence classes as linear extensions of explicit posets.
A combinatorial interpretation of the noncommutative inverse Kostka matrix
该研究提出无瑕非交换对称函数展开为完全齐次非交换对称函数的组合公式，引入GBPR图与隧道钩，推广至斜形状及整数序列索引的无瑕函数，并拓展了Campbell的带状分解结果。
Noncommutative Symmetric Systems over Associative Algebras
本文是一系列论文中的第一篇（[Z4]--[Z7]）上的可交换或非可交换变量微分算子代数上的{ \it ${\mathcal N}$ CS $(\text{noncommutative symmetric})$系统}（[Z4]）; Grossman-Larson霍普夫代数上的${\mathcal N}$ CS系统（[GL]，[F]）;以及它们与反演问题（[BCW]，[E4]）和NCSF的专门化（[Z5]，[Z7]）的联系和应用。本文受[GKLLRT]关于非交换对称函数（NCSF）的开创性工作的启发，首先在结合$\mathbb Q$ -代数上建立了{ \it ${\mathcal N}$ CS系统}的概念.然后我们证明了一般的${\mathcal N}$ CS系统、双代数或霍普夫代数上的${\mathcal N}$ CS系统以及由[GKLLRT]中某些NCSF的生成函数形成的泛${\mathcal N}$ CS系统的一些结果.最后，我们回顾了一些主要结果，这些结果将在下面的文章[Z4]，[Z6]和[Z7]中得到证明，作为本文一般讨论的一些支持例子。
Riemann-Hilbert problem and the discrete Bessel kernel
我们利用离散模拟的黎曼-希尔伯特问题的方法导出了描述对称群的泊松化Plancherel测度的离散Bessel核.要做到这一点，我们定义离散模拟物的黎曼-希尔伯特问题和可积积分算子，并表明计算的预解的离散可积算子可以减少到解决相应的离散黎曼-希尔伯特问题。我们还给出了一个例子，显式可解的经典特殊函数，当一个离散的黎曼-希尔伯特问题收敛在一定的缩放限制到一个传统的一个例子，起源于无限对称群的表示论。
Noncommutative Symmetric Functions and the Inversion Problem
设$K$是任何有单位的交换$\bQ$ -代数和$z=(z_1, z_2, ..., z_n)$交换或非交换变量.设$t$为形式中心参数，$\kttzz$为$z$在$K[[t]]$上的形式幂级数代数.在\cite{GTS-II}中，对于$\kttzz$与$H_{t=0}(z)=0$和$o(H(z))\geq 1$的每个自同构$F_t(z)=z-H_t(z)$，构造了一个\cNcs（非交换对称）系统（\cite{GTS-I}）$\Oft$。由此，我们从NCSF（非交换对称函数）的Hopf代数$\cNsf$（\cite{G-T}）得到了一个Hopf代数同态$\cSft: \cNsf \to \cDzz$.本文首先利用\cite{G-T}中导出的NCSF的一些恒等式和同态$\cSft$，给出了\cNcs系统$\Oft$中任意两个微分算子列之间的恒等式列表.其次，我们应用这些恒等式导出了$u(F_t)$和$u(F_t^{-1})$ $(u(z)\in \kttzz)$的Taylor级数展开式的$\Oft$系统中微分算子形式的公式，$F_t$的D-Log和形式流以及$F_t$逆映射的反演公式。最后，我们讨论了著名的雅可比猜想与NCSF的连接。
Matrix symmetric and quasi-symmetric functions and noncommutative representation theory
本文基于紧凑整数矩阵构造降阶代数的二维类似物，研究其双代数结构，并将其与 Chen 迭代积分签名推广中的双代数联系起来。
Differential Operator Specializations of Noncommutative Symmetric Functions
设$K$是任何有单位的交换$\mathbb Q$ -代数和$z=(z_1, ..., z_n)$交换或非交换自由变量.设$t$是一个形式参数，它与$z$和$K$的元素互换。我们用$\kzz$和$\kttzz$分别统一表示$z$在$K$和$K[[t]]$上的形式幂级数代数。对于任何$α\geq 1$，令$\cDazz$是由$\kzz$的微分算子生成的酉代数，这些微分算子至少使$z$的度增加$α-1$，而$ \ataz $是$\kttzz$与$o(H_t(z))\geq α$和$H_{t=0}(z)=0$的自同构群$F_t(z)=z-H_t(z)$。首先，对任何固定的$α\geq 1$和$F_t\in \ataz$，我们引入了$\kzz$的五个微分算子序列，并证明了它们的生成函数构成了微分代数$\cDazz$上的一个$\mathcal N$ CS（非交换对称）系统[Z4].因此，利用文献[GKLLRT]中首次引入的非交换对称函数（NCSF）的生成函数所构成的$\mathcal N$ CS系统的泛性质，我们得到了一族Hopf代数同态$\cS_{F_t}: {\mathcal N}Sym \to \cDazz$ $(F_t\in \ataz)$，当$F_t$满足一定条件时，这些同态也是保阶的.注意，上面的同态$\cS_{F_t}$也可以被看作是$\kzz$的微分算子对NCSF的特殊化。其次…
The machinery of noncommutative Schur functions provides a general tool for obtaining Schur expansions for combinatorially defined symmetric functions. We extend this approach to a wider class of symmetric functions, explore its strengths and limitations, and obtain new results on Schur positivity. We introduce combinatorial gadgets called switchboards, an adaptation of the D graphs of S. Assaf, and show how symmetric functions associated to them (which include LLT, Macdonald, Stanley, and stable Grothendieck polynomials) fit into the noncommutative Schur functions approach. This extends earli…
The theory of noncommutative Schur functions can be used to obtain positive combinatorial formulae for the Schur expansion of various classes of symmetric functions, as shown by Fomin and Greene. We develop a theory of noncommutative super Schur functions and use it to prove a positive combinatorial rule for the Kronecker coefficients where one of the partitions is a hook, recovering previous results of the two authors. This method also gives a precise connection between this rule and a heuristic for Kronecker coefficients first investigated by Lascoux.
We prove a Murnaghan-Nakayama rule for the noncommutative Schur functions introduced by Bessenrodt, Luoto and van Willigenburg. In other words, we give an explicit combinatorial formula for expanding the product of a noncommutative power sum symmetric function and a noncommutative Schur function in terms of noncommutative Schur functions. In direct analogy to the classical Murnaghan-Nakayama rule, the summands are computed using a noncommutative analogue of border strips, and have coefficients equal to 1 or -1 determined by the height of these border strips. The rule is proved by interpreting …
Schur positivity of the spiders $S(a,2,1)$ and $S(a,4,1)$ via noncommutative symmetric functions
本文通过非交换对称函数和带子展开，证明了蜘蛛图 $S(a,2,1)$ 与 $S(a,4,1)$ 对所有 $a\ge1$ 均为 Schur 正，完成了这两族的正性分类。
We give noncommutative versions of the Redfield-Pólya theorem in WSym, the algebra of word symmetric functions, and in other related combinatorial Hopf algebras.
Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic…
This article discusses the representation theory of noncommutative algebras reality-based algebras with positive degree map over their field of definition. When the standard basis contains exactly two nonreal elements, the main result expresses the noncommutative simple component as a generalized quaternion algebra over its field of definition. The field of real numbers will always be a splitting field for this algebra, but there are noncommutative table algebras of dimension $6$ with rational field of definition for which it is a division algebra. The approach has other applications, one of w…
Rooted trees and symmetric functions: Zhao's homomorphism and the commutative hexagon
最近微扰量子场论的工作导致了对Connes-Kreimer霍普夫代数的大量研究。它的（分次）对偶，有根树的Grossman-Larson霍普夫代数，已经被代数学家研究过了。L.福瓦西引入了Connes-Kreimer霍普夫代数的一个非交换版本，它是自对偶的。利用作者和W. Zhao，我们描述了一个交换图表，它将上述霍普夫代数相互联系起来，并与对称函数、非交换对称函数和拟对称函数的霍普夫代数联系起来。
A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion
Bogoliubov递推是微扰量子场论重整化过程中出现的一个特殊过程。它提供了收敛表达式，否则发散积分。我们在这里开发的非交换Rota-Baxter代数的功能标识的理论，这是编码，除其他外，这一过程中的背景下，康纳斯-Kreimer的霍普夫代数的重整化。我们的结果推广了交换Rota-Baxter代数的经典Spitzer型恒等式的Cartier-Rota理论。在经典的交换情形下，这些恒等式可以理解为从对称函数理论中导出。在这里，我们表明，一个类似的性质持有非交换Rota-Baxter代数。也就是说，我们表明，在非交换设置功能的身份可以从非交换对称函数的理论。李幂等元，特别是Dynkin幂等元在这个过程中起着至关重要的作用。他们的行动对亲幂幺群，如那些微扰重整化的详细描述的方式。
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