Frank Ingram, Naihuan Jing, Ernie Stitzinger
We systematically study wreath product Schur functions and give a combinatorial construction using colored partitions and tableaux. The Pieri rule and the Littlewood-Richardson rule are studied. We also discuss the connection with representations of generalized symmetric groups.
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We construct Hopf algebras whose elements are representations of combinatorial automorphism groups, by generalising a theorem of Zelevinsky on Hopf algebras of representations of wreath products. As an application we attach symmetric functions to representations of graph automorphism groups, generalising and refining Stanley's chromatic symmetric function.
A Gelafand model for wreath products $\Z_r\wr S_n$ is constructed. The proof relies on a combinatorial interpretation of the characters of the model, extending a classical result of Frobenius and Schur.
The symmetric group $S_{2n}$ and the hyperoctaheadral group $H_{n}$ is a Gelfand triple for an arbitrary linear representation $ϕ$ of $H_{n}$. Their $ϕ$-spherical functions can be caught as transition matrix between suitable symmetric functions and the power sums. We generalize this triplet in the term of wreath product. It is shown that our triplet are always to be a Gelfand triple. Furthermore we study the relation between their spherical functions and multi-partition version of the ring of symmetric functions.
We prove that if a finite group $H$ has a generalized involution model, as defined by Bump and Ginzburg, then the wreath product $H \wr S_n$ also has a generalized involution model. This extends the work of Baddeley concerning involution models for wreath products. As an application, we construct a Gelfand model for wreath products of the form $A \wr S_n$ with $A$ abelian, and give an alternate proof of a recent result due to Adin, Postnikov, and Roichman describing a particularly elegant Gelfand model for the wreath product $\ZZ_r \wr S_n$. We conclude by discussing some notable properties of…
Let G be a finite group acting on the finite set X such that the corresponding (complex) permutation representation is multiplicity free. There is a natural rank and order preserving action of the wreath product G~S_n on the generalized Boolean algebra B_X(n). We explicitly block diagonalize the commutant of this action.
A character identity which relates irreducible character values of the hyperoctahedral group $B_n$ to those of the symmetric group $S_{2n}$ was recently proved by Lübeck and Prasad. Their proof is algebraic and involves Lie theory. We present a short combinatorial proof of this identity, as well as a generalization to other wreath products.
The algebra of conjugacy classes of the wreath product of a finite group with the symmetric group
本文定义了$G$-部分置换概念，利用组合代数工具证明了有限群与对称群圈积代数中心的结构系数是具有非负整数系数的$n$的多项式，推广了Ivanov和Kerov方法。
We prove analogues for sub-posets of the Dowling lattices of the results of Calderbank, Hanlon, and Robinson on homology of sub-posets of the partition lattices. The technical tool used is the wreath product analogue of the tensor species of Joyal.
In [F. Caselli, Involutory reflection groups and their models, J. Algebra 24 (2010), 370--393] it is constructed a uniform Gelfand model for all non-exceptional irreducible complex reflection groups which are involutory. This model can be naturally decomposed into the direct sum of submodules indexed by symmetric conjugacy classes, and in this paper we present a simple combinatorial description of the irreducible decomposition of these submodules if the group is the wreath product of a cyclic group with a symmetric group. This is attained by showing that such decomposition is compatible with t…
In this paper, we extend to a non-consecutive case, the study of the pattern matching condition on the wreath product of the cyclic group and the symmetric group initiated by the authors in a previous paper. The main focus of our paper is (colored) patterns of length 2, although a number of enumerative results for longer patterns are also presented. A new non-trivial bijective interpretation for the Catalan numbers is found, in terms of simultaneously bi-avoiding two patterns in a wreath product.
We study some combinatorial properties of partial wreath $k$-th power of the semigroup $\mathcal{IS}_d$. In particular, we calculate its order, the number of idempotents and the number of D-classes.
The center of the wreath product of symmetric groups algebra
本文将两个对称群的圈积视为块置换群并研究其共轭类，证明了其中心结构系数的多项式性质，从而恢复并推广了关于对称群及超八面体群代数的已有结果。
We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products $Γ\wr\SG_n$ and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon's multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras …
We describe explicitly the algebraic structure of the Terwilliger algebra of wreath products of cyclic schemes.
A wreath product is a method to construct an association scheme from two association schemes. We determine the automorphism group of a wreath product. We show a known result that a wreath product is Schurian if and only if both components are Schurian, which yields large families of non-Schurian association schemes and non-Schurian $S$-rings. We also study iterated wreath products. Kernel schemes by Martin and Stinson are shown to be iterated wreath products of class-one association schemes. The iterated wreath products give examples of projective systems of non-Schurian association schemes, w…
We present several multi-variable generating functions for a new pattern matching condition on the wreath product of the cyclic group and the symmetric group. Our new pattern matching condition requires that the underlying permutations match in the usual sense of pattern matching for the symmetric group and that the corresponding sequence of signs match in the sense of words, rather than the exact equality of signs which has been previously studied. We produce the generating functions for the number of matches that occur in elements of the wreath product for any pattern of length 2 by applying…
Revisiting Foulkes characters of wreath products
本文利用圈积 Schur–Weyl–Sergeev 对偶的两种特殊化，重构并分解 Foulkes 特征标，证明其为块特征标锥的极射线，并借余不变量代数构造相应表示。
We give a full description of the algebraic structures of the Bose-Mesner algebra and Terwilliger algebra of the wreath product of one-class association schemes.
Probability measures on families of partitions related to harmonic analysis on big wreath products
本文构造了紧群与无限对称群的花环积的广义正则表示，并给出了其特征由称为z-测度的划分族上的概率测度决定的显式公式，从而描述了大型花环积的广义正则表示的谱测度。
A permutation class which is closed under pattern involvement may be described in terms of its basis. The wreath product construction X \wr Y of two permutation classes X and Y is also closed, and we investigate classes Y with the property that, for any finitely based class X, the wreath product X \wr Y is also finitely based.
Inspired by the definition of generalized wreath product of permutation groups, we define the generalized wreath product of graphs, containing the classical Cartesian and wreath product of graphs as particular cases. We prove that the generalized wreath product of Cayley graphs of finite groups is the Cayley graph of the generalized wreath product of the corresponding groups.
Subgroups of the symmetric group $S_n$ act on powers of chains $C^n$ by permuting coordinates, and induce automorphisms of the ordered sets $C^n$. The quotients defined are candidates for symmetric chain decompositions. We establish this for some families of groups in order to enlarge the collection of subgroups $G$ of the symmetric group $S_n$ for which the quotient $B_n/G$ obtained from the $G$-orbits on the Boolean lattice $B_n$ is a symmetric chain order. The methods are also used to provide an elementary proof that quotients of powers of SCOs by cyclic groups are SCOs.
Pólya enumeration, wreath product symmetric functions, and moduli spaces of curves
本文提出了一种基于缠绕积对称函数的框架，用于计算稳定曲线模空间的$S_n$-等价欧拉特征，并通过Adams操作和幂和扭曲建立其与普通对称函数的关系。
Gelfand pairs involving the wreath product of finite abelian groups with symmetric groups
本文推广了对称群的Gelfand对理论，证明了对于有限群G，其与对称群的圈积构成Gelfand对的充分必要条件是G为阿贝尔群。
We present methods of calculating statistics generating functions over the colored permutation groups, and generalizing known theorems from the symmetric groups to general colored permutations groups.
Zeros in The Character Table of The Wreath Product of The Symmetric Group
本文证明了对称群圈积的特征表中零的个数下界，与多重划分数相关，给出了渐近估计。
The wreath product of graphs is a graph composition inspired by the notion of wreath product of groups, with interesting connections with Geometric Group Theory and Probability. This paper is devoted to the description of some degree and distance-based invariants, of large interest in Chemical Graph Theory, for a wreath product of graphs. An explicit formula is obtained for the Zagreb indices, in terms of the Zagreb indices of the factor graphs. A detailed analysis of distances in a wreath product is performed, allowing to describe the antipodal graph and to provide a formula for the Wiener in…
Various statistics on wreath products are defined via canonical words, "colored" right to left minima and "colored" descents. It is shown that refined counts with respect to these statistics have nice recurrence formulas of binomial-Stirling type. These extended Stirling numbers determine (via matrix inversion) dual systems, which are also shown to have combinatorial realizations within the wreath product. The above setting also gives rise to MacMahon type equi-distribution theorem over subsets with prescribed statistics.
In 2013, Beck and Braun proved and generalized multiple identities involving permutation statistics via discrete geometry. Namely, they recognized the identities as specializations of integer point transform identities for certain polyhedral cones. They extended many of their proof techniques to obtain identities involving wreath products, but some identities were resistant to their proof attempts. In this article, we provide a geometric justification of one of these wreath product identities, which was first established by Biagioli and Zeng.
In 1997 Clarke et al. studied a $q$-analogue of Euler's difference table for $n!$ using a key bijection $Ψ$ on symmetric groups. In this paper we extend their results to the wreath product of a cyclic group with the symmetric group. In particular we obtain a new mahonian statistic \emph{fmaf} on wreath products. We also show that Foata and Han's two recent transformations on the symmetric groups provide indeed a factorization of $Ψ$.
We examine the conjugacy growth series of all wreath products of the finitary permutation groups $\text{Sym}(X)$ and $\text{Alt}(X)$ for an infinite set $X$. We determine their asymptotics, and we characterize the limiting behavior between the $\text{Alt}(X)$ and $\text{Sym}(X)$ wreath products. In particular, their ratios form a limit if and only if the dimension of the symmetric wreath product is twice the dimension of the alternating wreath product.
The automorphism group of the composition of graphs $G \circ H$ contains the wreath product $Aut(H) \wr Aut(G)$ of the automorphism groups of the corresponding graphs. The classical problem considered by Sabidussi and Hemminger was under what conditions $G \circ H$ has no other automorphisms. In this paper we deal with the converse. If the automorphism group of a graph (or a colored graph or digraph) is the wreath product $A \wr B$ of permutation groups, then the graph must be the result of the corresponding construction. The question we consider is whether $A$ and $B$ must be the automorphism…
A generating set for the wreath product $\ZZ_r \wr S_n$ which leads to a nicely behaved weak order is presented, and properties of the resulting order are studied.
The Hamilton-Waterloo problem is a problem of graph factorization. The Hamilton-Waterloo problem HWP$(H;m,n;α,β)$ asks for a $2$-factorization of $H$ containing $α$ $C_m$-factors and $β$ $C_n$-factors. In this paper, we almost completely solve the Hamilton-Waterloo problem on wreath product graph $C_m \wr K_{16}$ with $C_{16}$-factors and $C_m$-factors for an odd integer $m$.
We propose an analogue of Solomon's descent theory for the case of a wreath product G ~ S_n, where G is a finite abelian group. Our construction mixes a number of ingredients: Mantaci-Reutenauer algebras, Specht's theory for the representations of wreath products, Okada's extension to wreath products of the Robinson-Schensted correspondence, Poirier's quasisymmetric functions. We insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group.
Terwilliger algebras of generalized wreath products of association schemes
本文正式证明了对称结合方案的广义 wreath 积可适用于任意结合方案，并构建了该积结构的邻接代数与 Terwilliger 代数的不可约表示，完善了这一统一 wreath 积与直积的结构理论。
A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a definition and detailed examination of `Cartesian decompositions' of the permuted set, relating them to certain `Cartesian systemsof subgroups'. These concepts, and the bijective connections between them, are explored in greater generality, with specific future applications in mind.
We classify $\mathcal{R}$- and $\mathcal{L}$-cross-sections of partial wreath product of inverse semigroups. As a corollary, we get the description of $\mathcal{R}$- and $\mathcal{L}$-cross-sections of the semigroupof partial automorphisms of finite regular rooted tree and compute also the number of different $\mathcal{R}$- ($\mathcal{L}$-) cross-sections in this semigroup.
We give a partial solution a question of Grigorchuk, Nekrashevych, Sushchanskii and Šunik by giving an algorithm to test whether a finite state element of an infinite iterated (permutational) wreath product $\hat G = \mathbb Z/k\mathbb Z\wr \mathbb Z/k\mathbb Z\wr \mathbb Z/k\mathbb Z\wr >...$ of cyclic groups of order $n$ acts spherically transitively. We can also decide whether two finite state spherically transitive elements of $\hat G$ are conjugate. For general infinite iterated wreath products, an algorithm is presented to determine whether two finite state automorphisms have the same im…
Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. Given groups G and H acting on sets X and Y respectively, we characterize the distinguishing number of the wreath product of G and H in terms of the number of distinguishing colorings of X with respect to G and the disting…
Free Inhomogeneous Wreath Product of Quantum Groups
该研究引入紧矩阵量子群的自由非齐次圈积以推广已有自由圈积，提出借助极大双连通子图确定连通图量子自同构群的技术，并验证其适用于三类图，给出对应计算算法。
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