Hyperoctahedral Group Research Articles

The 𝐬-Eulerian polynomials have only real roots

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.

Equivalence Classes of Full-Dimensional $$0/1$$ 0 / 1 -Polytopes with Many Vertices

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.

Double Macdonald polynomials as the stable limit of Macdonald superpolynomials

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 $$m$$ m is sufficiently large: their decomposition in the monomial basis is then independent of $$m$$ 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 $$t$$ 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 $$q,t$$ q , t -Kostka coefficients associated to the double Macdonald polynomials are shown to be $$q,t$$ q , t -analogs of the dimensions of the irreducible representations of the hyperoctahedral group $$B_n$$ 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 $$(2n+1)^n$$ ( 2 n + 1 ) n , a formula again characteristic of the Coxeter group of type $$B_n$$ B n . Finally, as a side result, we obtain a simple identity involving products of four Littlewood-Richardson coefficients.

On the subgroup structure of the hyperoctahedral group in six dimensions.

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.

Signed Excedance Enumeration in the Hyperoctahedral group

Several signed excedance-like statistics have nice formulae or generating functions when summed over the symmetric group and over its subset of derangements. We give counterparts of some of these results when we sum over the hyperoctahedral group and its subset of derangements. Our results motivate us to define and derive attractive bivariate formulae which generalise some of these results for the symmetric group.

Counting signed permutations by their alternating runs

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.

Imprimitivity systems and lattices of normal subgroups in D-hyperoctahedral groups

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.

A Basis for the Diagonally Signed-Symmetric Polynomials

Let $n\ge 1$ be an integer and let $B_{n}$ denote the hyperoctahedral group of rank $n$. The group $B_{n}$ acts on the polynomial ring $Q[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]$ by signed permutations simultaneously on both of the sets of variables $x_{1},\dots,x_{n}$ and $y_{1},\dots,y_{n}.$ The invariant ring $M^{B_{n}}:=Q[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]^{B_{n}}$ is the ring of diagonally signed-symmetric polynomials. In this article, we provide an explicit free basis of $M^{B_{n}}$ as a module over the ring of symmetric polynomials on both of the sets of variables $x_{1}^{2},\dots, x^{2}_{n}$ and $y_{1}^{2},\dots, y^{2}_{n}$ using signed descent monomials.

Subgroup majorization

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 Rn, permutes and changes the signs of components.

A New Statistic on the Hyperoctahedral Groups

We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of type B), and give a conjectural formula for its signed distributions over arbitrary descent classes. The statistic is analogous to the classical Coxeter length function, and features a parity condition. For descent classes which are singletons the conjectured formula gives the Poincaré polynomials of the varieties of symmetric matrices of fixed rank.For several descent classes we prove the conjectural formula. For this we construct suitable supporting sets for the relevant generating functions. We prove cancellations on the complements of these supporting sets using suitably defined sign reversinginvolutions.

Euler–Mahonian statistics via polyhedral geometry

A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to a pair of such statistics is an Euler–Mahonian distribution, a bivariate polynomial encoding the statistics; such distributions often appear in rational bivariate generating-function identities. We use techniques from polyhedral geometry to establish new multivariate identities generalizing those giving rise to many of the known Euler–Mahonian distributions. The original bivariate identities are then specializations of these multivariate identities. As a consequence of these new techniques we obtain bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics.

The Bruhat−Chevalley Order on Involutions of the Hyperoctahedral Group and the Combinatorics of B-Orbit Closures

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.

The isometry groups of the hamming spaces of periodic sequences

We consider the Hamming space of periodic (0, 1)-sequences and a continual family of its subspaces defined as direct limits of finite Hamming spaces. These subspaces form a complete lattice under inclusion which is isomorphic to the lattice of supernatural numbers. We explicitly describe the isometry groups of these spaces. This involves certain constructions similar to the hyperoctahedral groups but accounting for additional structures on the underlying sets.

Representation stability for the cohomology of the pure string motion groups

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.

Decoding generalised hyperoctahedral groups and asymptotic analysis of correctible error patterns

We demonstrate a majority-logic decoding algorithm for decoding the generalised hyperoctahedral group $C_m \wr S_n$ when thought of as an error-correcting code. We also find the complexity of this decoding algorithm and compare it with that of another, more general, algorithm. Finally, we enumerate the number of error patterns exceeding the correction capability that can be successfully decoded by this algorithm, and analyse this asymptotically.

Brauer algebras of type C are cellularly stratified

AbstractIn a recent paper Cohen, Liu and Yu introduce the Brauer algebra of type C. We show that this algebra is an iterated inflation of hyperoctahedral groups, and that it is cellularly stratified. This allows us to give an indexing set of the standard modules, results on decomposition numbers, and the conditions under which the algebra is quasi-hereditary.

Quantum Symmetries and Strong Haagerup Inequalities

In this paper, we consider families of operators $\{x_r\}_{r \in \Lambda}$ in a tracial C$^\ast$-probability space $(\mathcal A, \phi)$, whose joint $\ast$-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups $\{H_n^+\}_{n \in \N}$. We prove a strong form of Haagerup's inequality for the non-self-adjoint operator algebra $\mathcal B$ generated by $\{x_r\}_{r \in \Lambda}$, which generalizes the strong Haagerup inequalities for $\ast$-free R-diagonal families obtained by Kemp-Speicher \cite{KeSp}. As an application of our result, we show that $\mathcal B$ always has the metric approximation property (MAP). We also apply our techniques to study the reduced C$^\ast$-algebra of the free unitary quantum group $U_n^+$. We show that the non-self-adjoint subalgebra $\mathcal B_n$ generated by the matrix elements of the fundamental corepresentation of $U_n^+$ has the MAP. Additionally, we prove a strong Haagerup inequality for $\mathcal B_n$, which improves on the estimates given by Vergnioux's property RD \cite{Ve}.

Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures

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 an extension of them, based on Jack polynomials.

Girth of pancake graphs

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 .

Quantum invariant families of matrices in free probability

We consider (self-adjoint) families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group. For the free orthogonal and hyperoctahedral groups, we obtain complete characterizations of the invariant families in terms of an operator-valued R-cyclicity condition. This is a surprising contrast with the Aldous–Hoover characterization of jointly exchangeable arrays.