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The Mallows measures on the symmetric group S n form a deformation of the uniform distribution. These measures are commonly used in mathematical statistics, and in recent years they found applications in other areas of mathematics as well. As shown by Gnedin and Olshanski, there exists an analog of the Mallows measures on the infinite symmetric group. These new measures are diffuse, and they are quasi-invariant with respect to the two-sided action of a countable dense subgroup. The purpose of the present note is to extend the Gnedin–Olshanski construction to the infinite hyperoctahedral group. Along the way, we obtain some results for the Mallows measures on finite hyperoctahedral groups, which may be of independent interest.

We prove that a totally complex algebraic number field K; having a conjugate which is not closed under complex conjugation, can be generated by a reciprocal integer, when the Galois group of its normal closure is contained in the hyperoctahedral group Bdeg(K)/2.Keywords: Reciprocal integers, unit primitive elements, reciprocal numbers

We study the joint distribution of descents and sign for elements of the symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and $B$). For both groups, this has an application to riffle shuffling: for large decks of cards the sign is close to random after a single shuffle. In both groups, we derive generating functions for the Eulerian distribution refined according to sign, and use them to give two proofs of central limit theorems for positive and negative Eulerian numbers., 25 pages

The hyperoctahedral group $H_N$ is known to have two natural liberations: the "good" one $H_N^+$, which is the quantum symmetry group of $N$ segments, and the "bad" one $\bar{O}_N$, which is the quantum symmetry group of the $N$-hypercube. We study here this phenomenon, in the general "quizzy" framework, which covers the various liberations and twists of $H_N,O_N$. Our results include: (1) an interpretation of the embedding $\bar{O}_N\subset S_{2^N}^+$, as corresponding to the antisymmetric representation of $O_N$, (2) a study of the liberations of $H_N$, notably with the result $=O_N^+$, and (3) a comparison of the $k$-orbitals for the inclusions $H_N\subset H_N^+$ and $H_N\subset\bar{O}_N$, for $k\in\mathbb N$ small., Comment: 34 pages

The algebra of symmetric functions is a major tool in algebraic combinatorics that plays a central role in the representation theory of the symmetric group. This thesis deals with quasisymmetric functions, a powerful generalisation introduced by Gessel in 1984, with significant applications in the enumeration of major combinatorial objects as permutations, Young tableaux and P-partitions. More specifically we find a new connection between Chow's extension of quasisymmetric functions to Coxeter groups of type B and domino tableaux. It allows us to contribute new results to various fields including the structure constants of type B Solomon's descent algebra, the extension of the theory of Schur-positivity to signed permutations and the study a q-deformed type B Cauchy formula with important implications regarding statistics for domino tableaux.Among the remarkable bases of the algebra of symmetric functions, Schur functions received a particular attention as they are strongly related to the irreducible characters of the general linear group and Young diagrams. The Schur symmetric function is the generating function for semistandard Young tableaux. This result extends to skew shapes and allows to write any (skew-) Schur function as the sum of Gessel's fundamental quasisymmetric functions indexed by the descent set of all standard Young tableaux of a given shape. Furthermore the celebrated Cauchy formula for Schur functions gives an algebraic proof of the Robinson-Schensted-Knuth correspondence. Finally, the structure constants for the outer product and inner product of Schur polynomials are respectively the Littlewood-Richardson and Kronecker coefficients, two important families of coefficients with various combinatorial and algebraic applications. Using known results about Gessel's fundamental quasisymmetric functions we show that these properties imply directly and in a pure algebraic fashion, various results for the structure constants of the Solomon descent algebra of a finite Coxeter group of type A and the descent preserving property of the Robinson-Schensted correspondence, an essential tool to identify Schur-positive sets, i.e. sets of permutations whose associated quasisymmetric function is symmetric and can be written as a non-negative sum of Schur symmetric functions.To extend these results to Coxeter groups of type B we introduced a family of modified generating functions for domino tableaux and relate it to Chow's type B fundamental quasisymmetric functions. Thanks to this relation we derive new formulas relating the structure constants of the type B Solomon's descent algebra with type B Kronecker and Littlewood-Richardson coefficients.It further allows us to introduce a new type B extension of Schur-positivity based on a definition of descent for signed permutations that is conform to the abstract definition of Solomon for any Coxeter groups. We design descent preserving bijections between signed arc permutations and sets of domino tableaux to show that they are indeed type B Schur-positive.Finally, we introduce a q-deformation of the modified generating functions for domino tableaux to extend a type B Cauchy identity by Lam and link it with Chow's quasisymmetric functions. We apply this result to a new framework of type B q-Schur positivity and to prove new equidistribution results for some sets of domino tableaux., L'algèbre des fonctions symétriques est un outil majeur de la combinatoire algébrique qui joue un rôle central dans la théorie des représentations du groupe symétrique. Cette thèse traite des fonctions quasisymétriques, une puissante généralisation introduite par Gessel en 1984, avec des applications significatives dans l'énumération d'objets combinatoires majeurs tels que les permutations, les tableaux de Young et les P-partitions. Plus précisément, nous trouvons un nouveau lien entre l'extension des fonctions quasisymétriques de Chow à des groupes de Coxeter de type B et des tableaux de dominos. Ceci nous permet d'apporter de nouveaux résultats dans divers domaines, notamment les constantes de structure de l'algèbre de descente de Solomon de type B, l'extension de la théorie de la Schur-positivité aux permutations signées et l'étude d'une formule de Cauchy de type Bq-déformée avec des implications importantes statistiques pour les tableaux dominos.Parmi les bases remarquables de l'algèbre des fonctions symétriques, les fonctions de Schur ont fait l'objet d'une attention particulière car elles sont étroitement liées aux caractères irréductibles du groupe linéaire général et aux diagrammes de Young. La fonction symétrique de Schur est la fonction génératrice des tableaux de Young semistandards. Ce résultat s'étend aux formes gauches et permet d'écrire n'importe quelle fonction de Schur (gauche) comme la somme des fonctions quasisymétriques fondamentales de Gessel, indexées par l'ensemble de descente de tous les tableaux de Young standard d'une forme donnée. En outre, la célèbre formule de Cauchy pour les fonctions de Schur donne une preuve algébrique de la correspondance de Robinson-Schensted-Knuth. Enfin, les constantes de structure pour la multiplication et la comultiplication des polynômes de Schur sont respectivement les coefficients de Littlewood-Richardson et de Kronecker, deux familles importantes de coefficients ayant diverses applications combinatoires et algébriques. En utilisant des résultats connus sur les fonctions quasisymétriques fondamentales de Gessel, nous montrons que ces propriétés impliquent directement et de façon purement algébrique divers résultats pour les constantes de structure de l'algèbre de descente de Salomon d'un groupe de Coxeter fini de type A et la propriété de préservation de descente de la correspondance de Robinson-Schensted, un outil essentiel pour identifier les ensembles Schur-positifs, c'est-à-dire les ensembles de permutations dont la fonction quasisymétrique associée est symétrique et qui peut s'écrire sous la forme d'une somme non négative de fonctions symétriques de Schur.Pour étendre ces résultats aux groupes de Coxeter de type B, nous avons introduit une famille de fonctions génératrices modifiées pour les tableaux de dominos et la relions aux fonctions quasisymétriques fondamentales de type B de Chow. Grâce à cette relation, nous obtenons de nouvelles formules reliant les constantes de structure de l'algèbre de descente de Salomon de type B aux coefficients de Kronecker et de Littlewood-Richardson de type B.Cela nous permet en outre d'introduire une nouvelle extension de type B de la Schur-positivité basée sur une définition de la descente pour les permutations signées, conforme à la définition abstraite de Solomon pour tous les groupes de Coxeter. Nous concevons des bijections préservant la descente entre des permutations d'arc signées et des ensembles de tableaux de dominos afin de montrer qu'ils sont bien type B Schur-positifs.Enfin, nous introduisons une q-déformation des fonctions génératrices modifiées pour les tableaux de dominos afin d'étendre une identité de Cauchy de type B proposée par Lam et de la lier aux fonctions quasisymétriques de Chow. Nous appliquons ce résultat à un nouveau cadre de positivité de type Bq-Schur et à la démonstration de nouveaux résultats d'équidistribution pour certains ensembles de tableaux de dominos.

We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the forms $(i,n)$ and $(-i,n)$ for $1\leq i\leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n\log n$. We also show that this shuffle exhibits the cutoff phenomenon. In the appendix, we show that a similar random walk on the demihyperoctahedral group $D_n$ also has a cutoff at $\left(n-\frac{1}{2}\right)\log n$., Comment: 21 pages, 2 figures, 1 table. Minor revisions. The main results are stated in the introduction. Remarks 2.5 and 4.2 are added. Theorem 2.1 is rephrased. Corrections are made in Section 3. The proof of the lower bound is simplified using probabilistic techniques, thanks to an anonymous referee of ALEA Lat. Am. J. Probab. Math. Stat. for the suggestion. Few minor changes are done in Theorem B.2

We relate character theory of the symmetric groups $S_{2n}$ and $S_{2n+1}$ with that of the hyperoctahedral group $B_n = ({\mathbb Z}/2)^n \rtimes S_n$, as part of the expectation that the character theory of reductive groups with diagram automorphism and their Weyl groups, is related to the character theory of the fixed subgroup of the diagram automorphism., Comment: This version contains an appendix by Arvind Ayyer

The multiplicity-free subgroups (strong Gelfand subgroups) of wreath products are investigated. Various useful reduction arguments are presented. In particular, we show that for every finite group [Formula: see text], the wreath product [Formula: see text], where [Formula: see text] is a Young subgroup, is multiplicity-free if and only if [Formula: see text] is a partition with at most two parts, the second part being 0, 1, or 2. Furthermore, we classify all multiplicity-free subgroups of hyperoctahedral groups. Along the way, we derive various decomposition formulas for the induced representations from some special subgroups of hyperoctahedral groups.

We have shown that powerful matrix-type-based multinomial generators together with combinatorial techniques can be applied to derive the conjugacy classes, irreducible representations and the character tables of n-dimensional hyperoctahedral or hypercube groups of order n! × 2n, which are generalisations of non-rigid molecular theory formulated by Longuet-Higgins. The hyperoctahedral groups appear in the symmetry groups of non-rigid molecules, non-rigid water clusters, and other weakly bound van der Waals complexes that exhibit rapid tunnelling motion among various low-lying surmountable potential energy minima. These groups also have applications in quantum chromodynamics in the classification of states of quarks. The associated combinatorics has applications in enumerations, nuclear spin statistics and NMR spectroscopy. Finally, the groups enhance understanding of chirality in biosystems, that is, the functionality of intrinsically disordered proteins. We have derived the character tables of hyp...

We characterise number fields without a unit primitive element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$, with degree $2d$ where $d$ is odd, and having a unit primitive element, can be generated by a reciprocal integer if and only if $K$ is not CM and the Galois group of the normal closure of $K$ is contained in the hyperoctahedral group $B_{d}$.

This paper discusses the asymptotic behaviour of the number of descents in a random signed permutation and its inverse, which was posed as an open problem by Chatterjee and Diaconis in a recent publication. For that purpose, we generalize their result for the asymptotic normality of the number of descents in a random permutation and its inverse to other finite reflection groups. This is achieved by applying their proof scheme on signed permutations, so elements of Coxeter groups of type $ B_n $, which is also known as the hyperoctahedral group. Furthermore, a similar central limit theorem for elements of Coxeter groups of type $D_n$ is derived via Slutsky's Theorem and a bound on the Wasserstein distance of certain normalized statistics with local dependency structures and bounded local components is proven for both types of Coxeter groups. In addition, we show a two-dimensional central limit theorem via the Cram\'er-Wold device., Comment: 9 pages

Theta-vexillary signed permutations are elements in the hyperoctahedral group that index certain classes of degeneracy loci of type B and C. These permutations are described using triples of $s$-tuples of integers subject to specific conditions. The objective of this work is to present different characterizations of theta-vexillary signed permutations, describing them in terms of corners in the Rothe diagram and pattern avoidance., Comment: 26 pages

We describe up to isomorphism all unitary hypergroups of order 3 over a group that arise from the dihedral group D9.

Arc permutations, which were originally introduced in the study of triangulations and characters, have recently been shown to have interesting combinatorial properties. The first part of this paper continues their study by providing signed enumeration formulas with respect to their descent set and major index. Next, we generalize the notion of arc permutations to the hyperoctahedral group in two different directions. We show that these extensions to type $B$ carry interesting analogues of the properties of type $A$ arc permutations, such as characterizations by pattern avoidance, and elegant unsigned and signed enumeration formulas with respect to the flag-major index., Comment: Version to appear in Journal of Combinatorics

In 1997 Bousquet-M\'elou and Eriksson introduced lecture hall partitions as the inversion vectors of elements of the parabolic quotient $\widetilde{C}/C$. We provide a new view of their correspondence that allows results in one domain to be translated into the other. We determine the equivalence between combinatorial statistics in each domain and use this correspondence to translate certain generating function formulas on lecture hall partitions to new observations about $\widetilde{C}/C$., Comment: 15 pages

We make a connection between the braid group and signed permutations. Using this link, we describe a commutative diagram which contains the fundamental sequence for the braid group.

We consider the classical Mahonian statistics on the set $B_n(\Sigma)$ of signed permutations in the hyperoctahedral group $B_n$ which avoid all patterns in $\Sigma$, where $\Sigma$ is a set of patterns of length two. In 2000, Simion gave the cardinality of $B_n(\Sigma)$ in the cases where $\Sigma$ contains either one or two patterns of length two and showed that $\left|B_n(\Sigma)\right|$ is constant whenever $\left|\Sigma\right|=1$, whereas in most but not all instances where $\left|\Sigma\right|=2$, $\left|B_n(\Sigma)\right|=(n+1)!$. We answer an open question of Simion by providing bijections from $B_n(\Sigma)$ to $S_{n+1}$ in these cases where $\left|B_n(\Sigma)\right|=(n+1)!$. In addition, we extend Simion's work by providing a combinatorial proof in the language of signed permutations for the major index on $B_n(21, \bar{2}\bar{1})$ and by giving the major index on $D_n(\Sigma)$ for $\Sigma =\{21, \bar{2}\bar{1}\}$ and $\Sigma=\{12,21\}$. The main result of this paper is to give the inversion generating functions for $B_n(\Sigma)$ for almost all sets $\Sigma$ with $\left|\Sigma\right|\leq2.$

We study the problem of determining the least symmetric triangle, which arises both from pure geometry and from the study of molecular chirality in chemistry. Using the correspondence between planar $n$-gons and points in the Grassmannian of 2-planes in real $n$-space introduced by Hausmann and Knutson, this corresponds to finding the point in the fundamental domain of the hyperoctahedral group action on the Grassmannian which is furthest from the boundary, which we compute exactly. We also determine the least symmetric obtuse and acute triangles. These calculations provide prototypes for computations on polygon and shape spaces., Comment: 17 pages, 12 figures

We study the roots of generalized Eulerian polynomials via a novel approach. We interpret Eulerian polynomials as the generating polynomials of a statistic over inversion sequences. Inversion sequences (also known as Lehmer codes or subexcedant functions) were recently generalized by Savage and Schuster, to arbitrary sequences s \mathbf {s} of positive integers, which they called s \mathbf {s} -inversion sequences. Our object of study is the generating polynomial of the ascent statistic over the set of s \mathbf {s} -inversion sequences of length n n . Since this ascent statistic over inversion sequences is equidistributed with the descent statistic over permutations, we call this generalized polynomial the s \mathbf {s} -Eulerian polynomial. The main result of this paper is that, for any sequence s \mathbf {s} of positive integers, the s \mathbf {s} -Eulerian polynomial has only real roots. This result is first shown to generalize several existing results about the real-rootedness of various Eulerian polynomials. We then show that it can be used to settle a conjecture of Brenti, that Eulerian polynomials for all finite Coxeter groups have only real roots, and partially settle a conjecture of Dilks, Petersen, Stembridge on type B affine Eulerian polynomials. It is then extended to several q q -analogs. We show that the MacMahon–Carlitz q q -Eulerian polynomial has only real roots whenever q q is a positive real number, confirming a conjecture of Chow and Gessel. The same holds true for the hyperoctahedral group and the wreath product groups, confirming further conjectures of Chow and Gessel, and Chow and Mansour, respectively. Our results have interesting geometric consequences as well.

Let $$Q_n$$Qn denote the $$n$$n-dimensional hypercube with vertex set $$V_n=\{0,1\}^n$$Vn={0,1}n. A $$0/1$$0/1-polytope of $$Q_n$$Qn is the convex hull of a subset of $$V_n$$Vn. This paper is concerned with the enumeration of equivalence classes of full-dimensional $$0/1$$0/1-polytopes under the symmetries of the hypercube. With the aid of a computer program, Aichholzer obtained the number of equivalence classes of full-dimensional $$0/1$$0/1-polytopes of $$Q_4$$Q4 and $$Q_5$$Q5 with any given number of vertices and those of $$Q_6$$Q6 up to 12 vertices. Let $$F_n(k)$$Fn(k) denote the number of equivalence classes of full-dimensional 0/1-polytopes of $$Q_n$$Qn with $$k$$k vertices. We present a method to compute $$F_n(k)$$Fn(k) for $$k>2^{n-2}$$k>2n-2. Let $$A_n(k)$$An(k) denote the number of equivalence classes of $$0/1$$0/1-polytopes of $$Q_n$$Qn with $$k$$k vertices, and let $$H_n(k)$$Hn(k) denote the number of equivalence classes of $$0/1$$0/1-polytopes of $$Q_n$$Qn with $$k$$k vertices that are not full-dimensional. So we have $$A_n(k)= F_n(k)+ H_n(k)$$An(k)=Fn(k)+Hn(k). It is known that $$A_n(k)$$An(k) can be computed by using the cycle index of the hyperoctahedral group. We show that for $$k>2^{n-2}$$k>2n-2, $$H_n(k)$$Hn(k) can be determined by the number of equivalence classes of $$0/1$$0/1-polytopes with $$k$$k vertices that are contained in every hyperplane spanned by a subset of $$V_n$$Vn. We also find a way to compute $$H_n(k)$$Hn(k) when $$k$$k is close to $$2^{n-2}$$2n-2. For the case of $$Q_6$$Q6, we can compute $$F_6(k)$$F6(k) for $$k>12$$k>12. Together with the computation of Aichholzer, we reach a complete solution to the enumeration of equivalence classes of full-dimensional 0/1-polytopes of $$Q_6$$Q6.

The subgroup structure of the hyperoctahedral group in six dimensions is studied, with particular attention to the subgroups isomorphic to the icosahedral group. The orthogonal crystallographic representations of the icosahedral group are classified, and their intersections are studied in some detail, using a combinatorial approach which involves results from graph theory and their spectra., The subgroup structure of the hyperoctahedral group in six dimensions is investigated. In particular, the subgroups isomorphic to the icosahedral group are studied. The orthogonal crystallographic representations of the icosahedral group are classified and their intersections and subgroups analysed, using results from graph theory and their spectra.

The purpose of this paper is to establish a connection between alternating runs of signed permutations in the hyperoctahedral group and left peaks of permutations in the symmetric group, and then to study some associated enumerative polynomials of signed permutations. Properties of these enumerative polynomials, including combinatorial formulas and recurrence relations are studied. In particular, we deduce a convolution formula connecting the number of alternating runs of permutations in the symmetric group to that in the hyperoctahedral group.

We study D-hyperoctahedral groups, diagonal inductive limits of hyperoctahedral groups. Also, we describe the Z 2-modules of periodic sequences over diagonal limits of symmetric groups under their action on the elements of a Z 2-module by permutation of coordinates. The imprimitivity systems for D-hyperoctahedral groups are characterized, and a full description of the lattices of their normal subgroups is given.

In this paper we consider cryptographic applications of the arithmetic on the hyperoctahedral group. On an appropriate subgroup of the latter, we particularly propose to construct public key cryptosystems based on the discrete logarithm. The fact that the group of signed permutations has rich properties provides fast and easy implementation and makes these systems resistant to attacks like the Pohlig-Hellman algorithm. The only negative point is that storing and transmitting permutations need large memory. Using together the hyperoctahedral enumeration system and what is called subexceedant functions, we define a one-to-one correspondence between natural numbers and signed permutations with which we label the message units.

The variation in genome arrangements among bacterial taxa is largely due to the process of inversion. Recent studies indicate that not all inversions are equally probable, suggesting, for instance, that shorter inversions are more frequent than longer, and those that move the terminus of replication are less probable than those that do not. Current methods for establishing the inversion distance between two bacterial genomes are unable to incorporate such information. In this paper we suggest a group-theoretic framework that in principle can take these constraints into account. In particular, we show that by lifting the problem from circular permutations to the affine symmetric group, the inversion distance can be found in polynomial time for a model in which inversions are restricted to acting on two regions. This requires the proof of new results in group theory, and suggests a vein of new combinatorial problems concerning permutation groups on which group theorists will be needed to collaborate with biologists. We apply the new method to inferring distances and phylogenies for published Yersinia pestis data., 19 pages, 7 figures, in Press, Journal of Mathematical Biology

Let G = Sp2n (ℂ) be the symplectic group, B be its Borel subgroup, and Φ = C n be the root system of G. To each involution σ in the Weyl group W of Φ, one can assign an orbit Ω σ of the coadjoint action of B on the dual space of the Lie algebra of the unipotent radical of B. Let σ, τ be involutions in W. It is proved that Ω σ is contained in the closure of Ω τ if and only if σ is less than or equal to τ with respect to the Bruhat–Chevalley order on W. Bibliography: 15 titles.

We give another proof, using tools from Geometric Invariant Theory, of a result due to S. Sam and A. Snowden in 2014, concerning the stability of Kro-necker coefficients. This result states that some sequences of Kronecker coefficients eventually stabilise, and our method gives a nice geometric bound from which the stabilisation occurs. We perform the explicit computation of such a bound on two examples, one being the classical case of Murnaghan's stability. Moreover, we see that our techniques apply to other coefficients arising in Representation Theory: namely to some plethysm coefficients and in the case of the tensor product of representations of the hyperoctahedral group., Manuscripta mathematica, Springer Verlag, In press, \&\#x3008;https://doi.org/10.1007/s00229-018-1021-4\&\#x3009

Centrosymmetric involutions in the symmetric group ${\mathcal S}_{2n}$ are permutations $\pi$ such that $\pi=\pi^{-1}$ and $\pi(i)+\pi(2n+1-i)=2n+1$ for all $i$ , and they are in bijection with involutions of the hyperoctahedral group. We describe the distribution of some natural descent statistics on $321$ -avoiding centrosymmetric involutions , including the number of descents in the first half of the involution, and the sum of the positions of these descents. Our results are based on two new bijections , one between centrosymmetric involutions in $ {\mathcal S} _{2n}$ and subsets of $\{1,\dots,n\}$ , and another one showing that certain statistics on Young diagrams that fit inside a rectangle are equidistributed . We also use the latter bijection to refine a known result stating that the distribution of the major index on $321$ -avoiding involutions is given by the $q$ -analogue of the central binomial coefficients.

Let F be an infinite field of characteristic different from 2 and G a torsion group. Let the group ring FG have an involution induced from an involution on G. We survey recent results establishing the conditions under which the unit group of FG satisfies a *-group identity. The surprising outcome is that the unit group satisfies a *-group identity if and only if the symmetric units of FG satisfy a group identity.

The cohomology of the pure string motion group PSigma_n admits a natural action by the hyperoctahedral group W_n. Church and Farb conjectured that for each k > 0, the sequence of degree k rational cohomology groups of PSigma_n is uniformly representation stable with respect to the induced action by W_n, that is, the description of the groups' decompositions into irreducible W_n representations stabilizes for n >> k. We use a characterization of the cohomology groups given by Jensen, McCammond, and Meier to prove this conjecture. Using a transfer argument, we further deduce that the rational cohomology groups of the string motion group vanish in positive degree. We also prove that the subgroup of orientation-preserving string motions, also known as the braid-permutation group, is rationally cohomologically stable in the classical sense., 24 pages, 3 figures

The convex hull of n+1 affinely independent vertices of the unit n-cube Cn is called a 0/1-simplex. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. In terms of linear algebra, acute 0/1-simplices in Cn can be described by nonsingular 0/1-matrices P of size n x n whose Gramians have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. The first part of this paper deals with giving a detailed description of how to efficiently compute, by means of a computer program, a representative from each orbit of an acute 0/1-simplex under the action of the hyperoctahedral group Bn of symmetries of Cn. A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature for n < 7. Using the computed cycle indices in combination with Polya's theory of enumeration shows that acute 0/1-simplices are extremely rare among all 0/1-simplices. In the second part of the paper, we study the 0/1-matrices that represent the acute 0/1-simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg 0/1-matrices of size n x n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler's Tree of Fractions that enumerates the rationals between 0 and 1. Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric matrices., 52 pages, 25 figures

We consider four families of pancake graphs, which are Cayley graphs, whose vertex sets are either the symmetric group on n objects or the hyperoctahedral group on n objects and whose generating sets are either all reversals or all reversals inverting the first k elements (called prefix reversals). We find that the girth of each family of pancake graphs remains constant after some small threshold value of n.

The symmetric group S 2 n and the hyperoctahedral group H n is a Gelfand triple for an arbitrary linear representation φ of H n . Their φ-spherical functions can be caught as a 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 is always a Gelfand triple. Furthermore we study the relation between their spherical functions and a multi-partition version of the ring of symmetric functions.

We generalize two bijections due to Garsia and Gessel to compute the generating functions of the two vector statistics (des(G), maj, l(G), col) and (des(G), ides(G), maj, imaj, col, icol) over the wreath product of a symmetric group by a cyclic group. Here desG, l(G), maj, col, idesG, imaj(G), and icol denote the number of descents, length, major index, color weight, inverse descents, inverse major index, and inverse color weight, respectively. Our main formulas generalize and unify several known identities due to Brenti, Carlitz, Chow-Gessel, Garsia-Gessel, and Reiner on various distributions of statistics over Coxeter groups of type A and B.

The final result of this article gives the order of the extension $$ 1 \longrightarrow P/[P,P]{\mathop{\longrightarrow} \limits^j} B/[P,P] {\mathop{\longrightarrow} \limits^p} W \longrightarrow 1$$ as an element of the cohomology group H 2(W, P/[P, P]) (where B and P stands for the braid group and the pure braid group associated to the complex reflection group W). To obtain this result, we first refine Stanley-Springer’s theorem on the abelianization of a reflection group to describe the abelianization of the stabilizer N H of a hyperplane H. The second step is to describe the abelianization of big subgroups of the braid group B of W. More precisely, we just need a group homomorphism from the inverse image of N H by p (where p: B → W is the canonical morphism) but a slight enhancement gives a complete description of the abelianization of p −1(W′) where W′ is a reflection subgroup of W or the stabilizer of a hyperplane. We also suggest a lifting construction for every element of the centralizer of a reflection in W.

We count derangements, involutions and unimodal elements in the wreath product Cr ≀ Sn by the numbers of excedances, fixed points and 2-cycles. Properties of the generating functions, including combinatorial formulas, recurrence relations and real-rootedness are studied. The results obtained specialize to those on the symmetric group Sn and on the hyperoctahedral group Bn when r = 1, 2, respectively.

We give some interpretations to certain integer sequences in terms of parameters on Grand-Dyck paths and coloured noncrossing partitions, and we find some new bijections relating Grand-Dyck paths and signed pattern avoiding permutations. Next we transfer a natural distributive lattice structure on Grand-Dyck paths to coloured noncrossing partitions and signed pattern avoiding permutations, thus showing, in particular, that it is isomorphic to the structure induced by the (strong) Bruhat order on a certain set of signed pattern avoiding permutations., 25 pages, submitted

Using non-rigid group theory, it was previously shown that the full non-rigid group of 2,4-dimethylbenzene is an ummatured and isomorphic to the group C2?(C3wrC2) of order 36, where Cn is the cyclic group of order n, the symbols ? and wr stand for the direct and wreath products, respectively. Herein, it is first shown that this group has 12 dominant classes. Then, the Markaracter Table, the Table of all integer-valued characters and the unit subduced cycle index (USCI) Table of the full non-rigid group of 2,4-dimethylbenzene are successfully derived for the first time.

We prove a result showing the connection between induced orthogonal representations and corestrictions of group extensions derived from their Clifford groups. By the means of the corestriction map we then obtain new obstructions to the μ p -embedding problems given by the group extensions of the modular p-group, and to one μ 2 -embedding problem given by a group extension of the dihedral group.

We define a type B derangement polynomial by q-counting derangements by the number of excedances in the hyperoctahedral group Bn. Properties of the derangement polynomial, including the recurrence relation, generating function, real-rootedness, reciprocity and a unimodal decomposition arising from a symmetric function identity, are studied. The results presented generalize and unify earlier ones due to Brenti, Stanley, Stembridge and Zhang.

Let A be an algebra over a field of characteristic zero with an additional structure of superalgebra or algebra with involution. The ordinary representation theory of the hyperoctahedral group \({\mathbb{Z}_2\wr S_n}\) is exploited in order to study the super-identities or the star-identities of A. One associates to A a sequence \({\chi_n^{\mathbb{Z}_2}(A), n=1,2,\ldots,}\,{\rm of}\,\mathbb{Z}_2\wr S_n\) -characters and one of the main objective of the theory is to determine their decomposition into irreducibles. Here we classify the super-identities and the star-identities in case the corresponding multiplicities are bounded by one. This is strictly related to the varieties of algebras whose lattice of subvarieties is distributive.

The abelianization of a group is its commutator quotient group. In this paper, we provide tables of the abelianizations of all the n-dimensional space groups for n = 1, 2, 3. We prove that the exponent of the torsion subgroup of the abelianization of an arbitrary n-dimensional space group Gamma divides the order of the point group of Gamma.

A combinatorial construction of a Gelafand model for the symmetric group and its Iwahori-Hecke algebra is presented., Comment: 15 pages, revised version, to appear in J. Algebra