Aaron Lauve, Mitja Mastnak
We identify a collection of primitive elements generating the Hopf algebra NCSym of symmetric functions in noncommuting variables and give a combinatorial formula for the antipode.
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
From a recent paper, we recall the Hopf monoid structure on the supercharacters of the unipotent uppertriangular groups over a finite field. We give cancelation free formula for the antipode applied to the bases of class functions and power sum functions, giving new cancelation free formulae for the standard Hopf algebra of supercharacters and symmetric functions in noncommuting variables. We also give partial results for the antipode on the supercharacter basis, and explicitly describe the primitives of this Hopf monoid.
In this paper, we give a cancellation-free antipode formula (for uniform matroids) for the restriction-contraction matroid Hopf algebra, using the technique of splitting and merging via a sign-reversing involution. The cancellation-free formula expresses the antipode of uniform matroids as a sum over certain ordered set partitions.
In this paper, we give a cancellation-free antipode formula for the matroid-minor Hopf algebra. We then explore applications of this formula. For example, the cancellation-free formula expresses the antipode of uniform matroids as a sum over certain ordered set partitions. We also prove that all matroids over any hyperfield (in the sense of Baker and Bowler) have cancellation-free antipode formulas; furthermore, the cancellations in the antipode are independent of the hyperfield structure and only depend on the underlying matroid.
We present a new formula for the antipode of incidence Hopf algebras. This formula is expressed as an alternating sum over forests. First, we prove the formula for incidence Hopf algebras of families of lattices by exhibiting a map from chains of a lattice to forests. Then, we extend the definition and present an analogous formula for the antipode of incidence Hopf algebras of families of posets. We characterize those families for which our formula is cancellation-free.
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…
The antipode of of a Com-PreLie Hopf algebra
研究Com-PreLie Hopf代数中对极与preLie积的相容性，以Connes-Kreimer Hopf代数为范例，揭示Connes-Moscovici子代数对极的组合系数可由迭代调和和计算。
The permutation pattern Hopf algebra is a commutative filtered and connected Hopf algebra. Its product structure stems from counting patterns of a permutation, interpreting the coefficients as permutation quasi-shuffles. The Hopf algebra was shown to be a free commutative algebra and to fit into a general framework of pattern Hopf algebras, via species with restrictions. In this paper we introduce the cancellation-free and grouping-free formula for the antipode of the permutation pattern Hopf algebra. To obtain this formula, we use the popular sign-reversing involution method, by Benedetti and…
We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.
If H is a connected, graded Hopf algebra, then Takeuchi's formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain cancellation-free formulas. We apply this idea to nine different examples. We rederive known formulas for the antipodes in the Hopf algebra of polynomials, the shuffle Hopf algebra, the Hopf algebra of quasisymmertic functions in both the monomial and fundamental bases, the Hopf algebra of multi-quasisymmetric functions in the fundamental basis, and the incidenc…
On the cancellation-free antipode formula for the Malvenuto-Reutenauer Hopf Algebra
本文针对 Malvenuto-Reutenauer Hopf 代数中一类以递减序列开头、递增序列结尾的置换，构造了无抵消的对极公式，并由此证明了 Benedetti 和 Sagan 提出的两个猜想。
Many combinatorial Hopf algebras $H$ in the literature are the functorial image of a linearized Hopf monoid $\bf H$. That is, $H={\mathcal K} ({\bf H})$ or $H=\overline{\mathcal K} ({\bf H})$. Unlike the functor $\overline{\mathcal K}$, the functor ${\mathcal K}$ applied to ${\bf H}$ may not preserve the antipode of ${\bf H}$. In this case, one needs to consider the larger Hopf monoid ${\bf L}\times{\bf H}$ to get $H={\mathcal K} ({\bf H})=\overline{\mathcal K}({\bf L}\times{\bf H})$ and study the antipode in ${\bf L}\times{\bf H}$. One of the main results in this paper provides a cancelation …
The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite simple graphs and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Takeuchi's and Schmitt's more general formulas for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial.
The Adams operators $Ψ_n$ on a Hopf algebra $H$ are the convolution powers of the identity of $H$. We study the Adams operators when $H$ is graded connected. They are also called Hopf powers or Sweedler powers. The main result is a complete description of the characteristic polynomial (both eigenvalues and their multiplicities) for the action of the operator $Ψ_n$ on each homogeneous component of $H$. The eigenvalues are powers of $n$. The multiplicities are independent of $n$, and in fact only depend on the dimension sequence of $H$. These results apply in particular to the antipode of $H$ (t…
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。
Hopf monoids of ordered simplicial complexes
本文通过引入Hopf类的概念，系统研究有序单纯复形与拟阵相关的Hopf幺半群结构，并在面初始复形和无界有序拟阵两类情形下计算了对极，得到无重数且无抵消的公式。
The peak algebra in noncommuting variables
本文将下降到峰映射与峰代数推广到非交换变量，定义 NCQSym、NCΠ 及非交换 Schur Q-函数，并证明其经典类似性质与广义 Dehn-Sommerville 方程。
Formulas for the Connes-Moscovici Hopf algebra
给出了Connes-Moscovici霍普夫代数$\mathcal{H}_{\tmop{CM}}$的余积和对极的显式表达式.要做到这一点，我们首先限制自己的一个子霍普夫代数$\mathcal{H}^1_{\tmop{CM}}$包含非平凡的元素，即那些的余积和对极是非平凡的。有两种方法可以获得显式公式。一方面，代数$\mathcal{H}^1_{\tmop{CM}}$同构于单位切同态群上坐标的Faà di Bruno霍普夫代数，并且使用替换自同构而不是自同构使计算变得容易。另一方面，在向量场的形式共轭和解析共轭的框架下，代数$\mathcal{H}^1_{\tmop{CM}}$同构于在重构理论中自然出现的经典shuffle霍普夫代数的子霍普夫代数.利用洗牌霍普夫代数的简单结构，我们再次导出了$\mathcal{H}^1_{\tmop{CM}}$中余积和对极的显式公式.
Three equivalent methods allow to compute the antipode of the Hopf algebras of Feynman diagrams in perturbative quantum field theory (QFT): the Dyson-Salam formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the first two hold generally for arbitrary connected graded Hopf algebras, the third one requires extra structure properties of the underlying Hopf algebra but has the nice property to reduce drastically the number of terms in the expression of the antipode (it is optimal in that sense).The present article is concerned with the forest formula: we show that it genera…
Prescribed duality dynamics in comodule categories
该研究构造出对极满而非双射且无来自双射对极Hopf代数非平凡同态的Hopf代数，解决了公开问题，并通过实现指定子空间格为余模格的现象给出构造，揭示了Hopf代数对极性质的对偶不对称性。
We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric…
We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on $c$-clusters in the theory of cluster algebras.
On the square of the antipode in a connected filtered Hopf algebra
本文推广了连通分次 Hopf 代数中反极元平方的幂零性结果，在更一般的余代数框架下证明指数可降低的条件，并给出充分条件。
Generalized permutahedra are a family of polytopes with a rich combinatorial structure and strong connections to optimization. We prove that they are the universal family of polyhedra with a certain Hopf algebraic structure. Their antipode is remarkably simple: the antipode of a polytope is the alternating sum of its faces. Our construction provides a unifying framework to organize numerous combinatorial structures, including graphs, matroids, posets, set partitions, linear graphs, hypergraphs, simplicial complexes, building sets, and simple graphs. We highlight three applications: 1. We obtai…
The extra basis in noncommuting variables
本文利用 Hopf 幺半群理论和 Fock 函子，给出了非交换变量对称函数 Hopf 代数 NCSym 中 x-基余积的组合描述，并发现了对称函数代数的一个新的乘法基。
In this work a combinatorial description is provided of a Faa di Bruno type Hopf algebra which naturally appears in the context of Fliess operators in nonlinear feedback control theory. It is a connected graded commutative and non-cocommutative Hopf algebra defined on rooted circle trees. A cancellation free forest formula for its antipode is given.
On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra
本文通过引入post-Hopf代数概念，研究post-Lie代数的泛包络代数的次邻接Hopf代数，利用扭结post-Hopf乘积给出组合反演公式，并得到Oudom-Guin同构的闭逆公式，进而导出有序树Grossman-Larson Hopf代数的无消去反演公式。
Combinatorial Hopf algebras give a linear algebraic structure to infinite families of combinatorial objects, a technique further enriched by the categorification of these structure via the representation theory of families of algebras. This paper examines a fundamental construction in group theory, the direct product, and how it can be used to build representation theoretic Hopf algebras out of towers of groups. A key special case gives us the noncommutative symmetric functions NSym, but there are many things that we can say for the general Hopf algebras, including the structure of their chara…
A study of nil Hecke algebras via Hopf algebroids
本文通过 Kostant 和 Kumar 定义的余乘法，利用 Demazure 算子与单反射间的混合二面体辫关系，将与 Coxeter 系统关联的仿射 nil Hecke 代数转化为无对极的 Hopf algebroid，并附带了相关领域的实例。
The supercharacter theory of algebra groups gave us a representation theoretic realization of the Hopf algebra of symmetric functions in noncommuting variables. The underlying representation theoretic framework comes equipped with two canonical bases, one of which was completely new in terms of symmetric functions. This paper simultaneously generalizes this Hopf structure by considering a larger class of groups while also restricting the representation theory to a more combinatorially tractable one. Using the normal lattice supercharacter theory of pattern groups, we not only gain a third cano…
Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Faà di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants…
Zimmermann Type Cancellation in the Free Faa di Bruno Algebra
由Brouder，Fabretti，and Krattenaler（BFK）在非交换Lagrange反演的上下文中引入的N-变量霍普夫代数可以与N-着色区间划分的关联代数的逆相一致。（BFK）对极及其反射决定了具有N个非交换变量的非交换系数幂级数的（通常不同的）左逆和右逆。在Faa di Bruno霍普夫代数的情况下，有一个类似的齐默尔曼取消公式。（BFK）对极的和数可以用压缩平面树上顶点的深度优先排序来索引，这同样适用于区间划分对极。这两个也可以索引的宽度第一顺序的顶点在无序的可收缩平面树，其中恰好一个非退化顶点出现在每一个级别。
Structure of the Malvenuto-Reutenauer Hopf algebra of permutations (Extended Abstract)
我们详细分析了置换的Malvenuto-Schottenauer霍普夫代数的结构。给出了它的对极的显式公式，证明了它是余自由余代数，确定了它的本原元和余根滤子，并证明了它可分解为拟对称函数的霍普夫代数上的交叉积.我们还描述了乘法的结构常数作为置换面体的一定数量的小平面。我们的结果揭示了这种霍普夫代数的结构与对称群上的弱序之间的密切关系。
We introduce the bicategory of bialgebras with coverings (which can be thought of as coalgebra-indexed families of morphisms), and provide a motivating application to the transfer of formulas for primitives and antipode. Additionally, we study properties of this bicategory and various sub-bicategories, and describe some universal constructions. Finally, we generalize Nichols' result on bialgebra quotients of Hopf algebra, which gives conditions on when the resulting bialgebra quotient is a Hopf algebra.
This is a short review on the Faà di Bruno formulas, and some Hopf algebras associated to them. These allow, among several other things, a short proof of the Lie-Scheffers theorem, and relating the Lagrange inversion formulas with antipodes. The link of Faà di Bruno formulas with the theory of set partitions is developed in some detail.
A universal framework for Hopf algebras in combinatorics
该研究通过对偶化组合霍普夫代数理论建立根霍普夫代数范畴，将Schur等函数扩展到根霍普夫代数，并应用于非交换变量对称函数，为组合对象提供统一理解。
We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these combinatorial Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanley's $(-1)$-color theorem. A $q$-analog version of this family of characters is also studied.
Structure of the Loday-Ronco Hopf algebra of trees
Loday和Ronco在平面二叉树集的线性跨度上定义了一个有趣的霍普夫代数结构。他们证明了包含非交换对称函数的霍普夫代数通过其树的霍普夫代数的置换因子在Malvenuto-Albertenauer霍普夫代数中的作用，并且这些映射对应于从对称群上的弱序到平面二叉树上的Tamari序到布尔代数的自然映射.我们进一步研究了树的霍普夫代数的结构，给出了树的一个新基，并利用这个基描述了树的乘积、余积和对极，从而阐明了树的霍普夫代数结构.证明了它与Foissy的非交换Connes-Kreimer霍普夫代数的同构，并证明了该代数与非交换对称函数的同构性，就像（交换）Connes-Kreimer霍普夫代数与对称函数的同构性一样.
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。