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In this paper, we study the determinant of irreducible representations of the generalized symmetric groups Z r ≀ S n . We give an explicit formula to compute the determinant of an irreducible representation of Z r ≀ S n . Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, given an integer n, r an odd prime and ζ a nontrivial multiplicative character of Z r ≀ S n with n < r, we obtain an explicit formula to compute N ζ (n) , the number of irreducible representations of Z r ≀ S n whose determinant is ζ. [ABSTRACT FROM AUTHOR]

In this paper, we study the determinant of irreducible representations of the generalized symmetric groups Z r ≀ S n . We give an explicit formula to compute the determinant of an irreducible representation of Z r ≀ S n . Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, given an integer n, r an odd prime and ζ a nontrivial multiplicative character of Z r ≀ S n with n < r, we obtain an explicit formula to compute N ζ (n) , the number of irreducible representations of Z r ≀ S n whose determinant is ζ. [ABSTRACT FROM AUTHOR]

For a graph with edge ordering, a linear order on the edge set, we obtain a permutation of vertices by considering the edges as transpositions of endvertices. It is known from D'enes' results that the permutation of a tree is a full cyclic for any edge ordering. As a corollary, D'enes counted up the number of representations of a full cyclic permutation by means of product of the minimal number of transpositions. Moreover, a graph with an edge ordering which the permutation is a full cyclic is characterized by graph embedding. In this article, we consider an analogy of these results for signed graphs and hyperoctahedral groups. We give a necessary and sufficient condition for a signed graph to have an edge ordering such that the permutation is an even (or odd) full cyclic. We show that the edge ordering of the signed tree with some loops always gives an even (or odd) full cyclic permutation and count up the number of representations of an odd full cyclic permutation by means of product of the minimal number of transpositions. [ABSTRACT FROM AUTHOR]

Let J n B denote the set of all fixed-point free involutions of the hyperoctahedral group B n , and let des B (π) denote the number of descents of the permutation π ∈ B n . We show that J n B (t) : = ∑ π ∈ J n B t des B (π) is symmetric, unimodal and γ -positive for n ≥ 2 . [ABSTRACT FROM AUTHOR]

We give a combinatorial description for the weak order on the hyperoctahedral group. This characterization is then used to analyze the order-theoretic properties of the shifted products of hyperoctahedral groups. It is shown that each shifted product is a disjoint union of some intervals, which can be convex embedded into a larger hyperoctahedral group. As an application, we investigate the monomial basis for the Hopf algebra H Sym of signed permutations, related to the fundamental basis via Möbius inversion on the weak order on hyperoctahedral groups. It turns out that the image of a monomial basis element under the descent map from H Sym to the algebra of type B quasi-symmetric functions is either zero or a monomial quasi-symmetric function of type B. [ABSTRACT FROM AUTHOR]

Let I n B denote the set of the involutions of the hyperoctahedral group B n , and let des B (π) denote the number of descents of the permutation π ∈ B n . We settle a problem of Moustakas which states that I n B (t) : = ∑ π ∈ I n B t des B (π) is γ -positive for n ≥ 1 . [ABSTRACT FROM AUTHOR]

We show that the ℋ n − 1 -conjugacy classes of ℋ n , where ℋ n is the hyperoctahedral group on 2 n elements, are indexed by marked bipartitions of n. This will lead us to prove that (ℋ n × ℋ n − 1 , diag (ℋ n − 1)) is a symmetric Gelfand pair and that the induced representation 1 diag (ℋ n − 1) ℋ n × ℋ n − 1 is multiplicity free. [ABSTRACT FROM AUTHOR]

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 F , the wreath product F ≀ S λ , where S λ is a Young subgroup, is multiplicity-free if and only if λ 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. [ABSTRACT FROM AUTHOR]

We consider a random walk on the hyperoctahedral group Bn generated by the signed permutations of the forms (i,n) and (-i,n) for 1≤i≤n. We call this the flip-transpose top with random shuffle on Bn. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order nlogn. We also show that this shuffle exhibits the cutoff phenomenon. In the appendix, we show that a similar random walk on the demihyperoctahedral group Dn also has a cutoff at (n-1/2)logn. [ABSTRACT FROM AUTHOR]

The convex hull of n + 1 affinely independent vertices of the unit n-cube In 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 In can be described by nonsingular 0/1-matrices P of size n × n whose Gramians G = PTP have an inverse that is strictly diagonally dominant, with negative off-diagonal entries [6, 7]. 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 [17] of symmetries of In. 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 [11] for n ≤ 6. Using the computed cycle indices for B3, . . . ,B11 in combination with Pólya’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 × n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem [4]. 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 [1, 24] that enumerates ℚ ⋂ (0, 1). Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric [14, 26] matrices.

The Eulerian distribution on the involutions of the symmetric group is unimodal, as shown by Guo and Zeng. In this paper we prove that the Eulerian distribution on the involutions of the hyperoctahedral group, when viewed as a colored permutation group, is unimodal in a similar way and we compute its generating function, using signed quasisymmetric functions. [ABSTRACT FROM AUTHOR]

Introduced by Solomon in his 1976 paper, the descent algebra of a finite Coxeter group received significant attention over the past decades. As proved by Gessel, in the case of the symmetric group its structure constants give the comultiplication table for the fundamental basis of quasisymmetric functions. We show that this latter property actually implies several well known relations linked to the Robinson–Schensted–Knuth correspondence and some of its generalisations. This provides a new link between these results and the theory of quasisymmetric functions and allows to derive more advanced formulae involving Kronecker coefficients. Using the theory of type B quasisymmetric functions introduced by Chow, we extend this connection to the hyperoctahedral group and derive new formulae relating the structure constants of the descent algebra of type B, the numbers of domino tableaux of given descent set and the Kronecker coefficients of the hyperoctahedral group. [ABSTRACT FROM AUTHOR]

The descent algebra of a finite Coxeter group W is a subalgebra of the group algebra defined by Solomon. Descent algebras of symmetric groups have properties that are not shared by other Coxeter groups. For instance, the natural map from the descent algebra of a symmetric group to its character ring is a surjection with kernel equal the Jacobson radical. Thus, the descent algebra implicitly encodes information about the representations of the symmetric group, and a complete set of primitive idempotents in the character ring leads to a decomposition of the group algebra into a sum of right ideals indexed by partitions. Stanley asked whether this decomposition of the regular representation of a symmetric group could be realized as a sum of representations induced from linear characters of centralizers. This question was answered positively by Bergeron, Bergeron, and Garsia, using a connection with the free Lie algebra on n letters, and independently by Douglass, Pfeiffer, and Röhrle, who connected the decomposition with the configuration space of n-tuples of distinct complex numbers. The Mantaci-Reutenauer algebra of a hyperoctahedral group is a subalgebra of the group algebra that contains the descent algebra. Bonnafé and Hohlweg showed that the natural map from the Mantaci-Reutenauer algebra to the character ring is a surjection with kernel equal the Jacobson radical. In 2008, Bonnafé asked whether the analog to Stanley’s question about the decomposition of the group algebra into a sum of induced linear characters holds. In this paper, we give a positive answer to Bonnafé’s question by explicitly constructing the required linear characters. [ABSTRACT FROM AUTHOR]

The n-dimensional hyperoctahedral group, denoted by Tn, is the group of the distance-preserving transforms of the n-dimensional cube. In this group, a norm and a deviation of an element can be defined as the maximum and minimum of the distances of each vertex and its transformed one. In previous papers the author proved results on the computation of these values and characterized the image of the norm and deviation functions. In the present paper we deal with the following problem: which ordered pair (p; q) belongs to a transform u so that the norm and the deviation of u is equal to p and q, respectively. [ABSTRACT FROM AUTHOR]

Introduced by Solomon in his 1976 paper, the descent algebra of a finite Coxeter group received significant attention over the past decades. As proved by Gessel, in the case of the symmetric group its structure constants give the comultiplication table for the fundamental basis of quasisymmetric functions. We show that this latter property actually implies several well known relations linked to the Robinson-Schensted-Knuth correspondence and some of its generalisations. This provides a new link between these results and the theory of quasisymmetric functions and allows to derive more advanced formulas involving Kronecker coefficients. Furthermore, using the theory of type B quasisymmetric functions introduced by Chow, we can extend our method to the hyperoctahedral group and derive some new formulas. [ABSTRACT FROM AUTHOR]

In this article we investigate the lattices of Dyck paths of type A and B under dominance order, and explicitly describe their Heyting algebra structure. This means that each Dyck path of either type has a relative pseudocomplement with respect to some other Dyck path of the same type. While the proof that this lattice forms a Heyting algebra is quite straightforward, the explicit computation of the relative pseudocomplements using the lattice-theoretic definition is quite tedious. We give a combinatorial description of the Heyting algebra operations join, meet, and relative pseudocomplement in terms of height sequences, and we use these results to derive formulas for pseudocomplements and to characterize the regular elements in these lattices. [ABSTRACT FROM AUTHOR]

A general setting to study a certain type of formulas, expressing characters of the symmetric group S n explicitly in terms of descent sets of combinatorial objects, has been developed by two of the authors. This theory is further investigated in this paper and extended to the hyperoctahedral group B n . Key ingredients are a new formula for the irreducible characters of B n , the signed quasisymmetric functions introduced by Poirier, and a new family of matrices of Walsh–Hadamard type. Applications include formulas for natural B n -actions on coinvariant and exterior algebras and on the top homology of a certain poset in terms of the combinatorics of various classes of signed permutations, as well as a B n -analogue of an equidistribution theorem of Désarménien and Wachs. [ABSTRACT FROM AUTHOR]

In this paper we use generating function methods to obtain new asymptotic results about spaces of F-stable maximal tori in GLn(Fq), Sp2n(Fq), and SO2n+1(Fq). We recover stability results of Church-Ellenberg-Farb and Jiménez Rolland-Wilson for "polynomial" statistics on these spaces, and we compute explicit formulas for their stable values. We derive a double generating function for the characters of the cohomology of ag varieties in type B/C, which we use to obtain analogs in type B/C of results of Chen: we recover "twisted homological stability" for the spaces of maximal tori in Sp2n(C) and SO2n+1(C), and we compute a generating function for their "stable twisted Betti numbers". We also give a new proof of a result of Lehrer using symmetric function theory. [ABSTRACT FROM AUTHOR]

We introduce a two-parameter function ϕ q + , q − on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length. We show that this signed reflection function ϕ q + , q − is positive definite if and only if it is an extreme character of the infinite hyperoctahedral group and we classify the corresponding set of parameters q + , q −. We construct the corresponding representations through a natural action of the hyperoctahedral group B (n) on the tensor product of n copies of a vector space, which gives a two-parameter analog of the classical construction of Schur–Weyl. We apply our classification to construct a cyclic Fock space of type B generalizing the one-parameter construction in type A found previously by Bożejko and Guta. We also construct a new Gaussian operator acting on the cyclic Fock space of type B and we relate its moments with the Askey–Wimp–Kerov distribution by using the notion of cycles on pair-partitions, which we introduce here. Finally, we explain how to solve the analogous problem for the Coxeter groups of type D by using our main result. [ABSTRACT FROM AUTHOR]

Macdonald superpolynomials provide a remarkably rich generalization of the usual Macdonald polynomials. The starting point of this work is the observation of a previously unnoticed stability property of the Macdonald superpolynomials when the fermionic sector $$m$$ is sufficiently large: their decomposition in the monomial basis is then independent of $$m$$ . These stable superpolynomials are readily mapped into bisymmetric polynomials, an operation that spoils the ring structure but drastically simplifies the associated vector space. Our main result is a factorization of the (stable) bisymmetric Macdonald polynomials, called double Macdonald polynomials and indexed by pairs of partitions, into a product of Macdonald polynomials (albeit subject to non-trivial plethystic transformations). As an off-shoot, we note that, after multiplication by a $$t$$ -Vandermonde determinant, this provides explicit formulas for a large class of Macdonald polynomials with prescribed symmetry. The factorization of the double Macdonald polynomials leads immediately to the generalization of basically every elementary properties of the Macdonald polynomials to the double case (norm, kernel, duality, evaluation, positivity, etc). When lifted back to superspace, this validates various previously formulated conjectures in the stable regime. The $$q,t$$ -Kostka coefficients associated to the double Macdonald polynomials are shown to be $$q,t$$ -analogs of the dimensions of the irreducible representations of the hyperoctahedral group $$B_n$$ . Moreover, a Nabla operator on the double Macdonald polynomials is defined, and its action on a certain bisymmetric Schur function can be interpreted as the Frobenius series of a bigraded module of dimension $$(2n+1)^n$$ , a formula again characteristic of the Coxeter group of type $$B_n$$ . Finally, as a side result, we obtain a simple identity involving products of four Littlewood-Richardson coefficients. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: The extension of majorization (also called the rearrangement ordering), to more general groups than the symmetric (permutation) group, is referred to as G-majorization. There are strong results in the case that G is a reflection group and this paper builds on this theory in the direction of subgroups, normal subgroups, quotient groups and extensions. The implications for fundamental cones and order-preserving functions are studied. The main example considered is the hyperoctahedral group, which, acting on a vector in , permutes and changes the signs of components. [Copyright &y& Elsevier]

We study D-hyperoctahedral groups, diagonal inductive limits of hyperoctahedral groups. Also, we describe the Z-modules of periodic sequences over diagonal limits of symmetric groups under their action on the elements of a Z-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. [ABSTRACT FROM AUTHOR]

A combinatorial construction of Gelfand models for the symmetric group, for its Iwahori-Hecke algebra and for the hyperoctahedral group is presented.

We study symmetric polynomials whose variables are odd-numbered Jucys-Murphy elements. They define elements of the Hecke algebra associated to the Gelfand pair of the symmetric group with the hyperoctahedral group. We evaluate their expansions in zonal spherical functions and in double coset sums. These evaluations are related to integrals of polynomial functions over orthogonal groups. Furthermore, we give their extension based on Jack polynomials. [ABSTRACT FROM AUTHOR]

The absolute order on the hyperoctahedral group B is investigated. Using a poset fiber theorem, it is proved that the order ideal of this poset generated by the Coxeter elements is homotopy Cohen-Macaulay. This method results in a new proof of Cohen-Macaulayness of the absolute order on the symmetric group. Moreover, it is shown that every closed interval in the absolute order on B is shellable and an example of a non-Cohen-Macaulay interval in the absolute order on D is given. Finally, the closed intervals in the absolute order on B and D which are lattices are characterized and some of their important enumerative invariants are computed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: Adin, Brenti, and Roichman [R.M. Adin, F. Brenti, Y. Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. in Appl. Math. 27 (2001) 210–224], in answering a question posed by Foata, introduced two descent numbers and major indices for the hyperoctahedral group , whose joint distribution generalizes an identity due to MacMahon and Carlitz. We shall show that yet another pair of statistics exists, and whose joint distribution constitutes a “natural” solution to Foata''s problem. [Copyright &y& Elsevier]