Hyperoctahedral Group Research Articles

The Eulerian Distribution on the Fixed-Point Free Involutions of the Hyperoctahedral Group Under the Natural Order

The Eulerian Distribution on the Fixed-Point Free Involutions of the Hyperoctahedral Group Under the Natural Order

The poset of Specht ideals for hyperoctahedral groups

The poset of Specht ideals for hyperoctahedral groups

Paired 2-disjoint path covers of burnt pancake graphs with faulty elements

The burnt pancake graph BPn is the Cayley graph of the hyperoctahedral group using prefix reversals as generators. Let {u,v} and {x,y} be any two pairs of distinct vertices of BPn for n≥4. We show that there are u−v and x−y paths whose vertices partition the vertex set of BPn even if BPn has up to n−4 faulty elements. On the other hand, for every n≥3 there is a set of n−2 faulty edges or faulty vertices for which such a fault-free disjoint path cover does not exist.

Signed graphs and signed cycles of hyperoctahedral groups

Signed graphs and signed cycles of hyperoctahedral groups

Wachs permutations, Bruhat order and weak order

We study the partial orders induced on Wachs and signed Wachs permutations by the Bruhat and weak orders of the symmetric and hyperoctahedral groups. We show that these orders are graded, determine their rank function, characterize their ordering and covering relations, and compute their characteristic polynomials, when partially ordered by Bruhat order, and determine their structure explicitly when partially ordered by right weak order.

On Derangement Polynomials of Type $D$

Enumeration of derangements in the symmetric group $\mathfrak{S}_n$ is classical. Extensions of the enumerative results to the hyperoctahedral group $B_n$ are combinatorially sound. That in the even-signed permutation group $D_n$ remains largely unexplored. Let $d_n^D(q) = \sum_{\sigma \in \mathcal{D}_n^D} q^{\operatorname{maj}(\sigma)}$ be the generating function of derangements in $D_n$ by their major indices. We study in this work properties of $d_n^D(q)$, including recurrence relations and factorial generating function. By proving the ratio monotonicity of $d_n^D(q)$, the unimodality, log-concavity and spiral property of $d_n^D(q)$ are also established.

The Double Fock Space of Type B

In this article, we introduce the notion of a double Fock space of type B. We will show that this new construction is compatible with combinatorics of counting positive and negative inversions on a hyperoctahedral group.

Recursive Symmetries: Chemically Induced Combinatorics of Colorings of Hyperplanes of an 8-Cube for All Irreducible Representations

We outline symmetry-based combinatorial and computational techniques to enumerate the colorings of all the hyperplanes (q = 1–8) of the 8-dimensional hypercube (8-cube) and for all 185 irreducible representations (IRs) of the 8-dimensional hyperoctahedral group, which contains 10,321,920 symmetry operations. The combinatorial techniques invoke the Möbius inversion method in conjunction with the generalized character cycle indices for all 185 IRs to obtain the generating functions for the colorings of eight kinds of hyperplanes of the 8-cube, such as vertices, edges, faces, cells, tesseracts, and hepteracts. We provide the computed tables for the colorings of all the hyperplanes of the 8-cube. We also show that the developed techniques have a number of chemical, biological, chiral, and other applications that make use of such recursive symmetries.

Topological Indices, Graph Spectra, Entropies, Laplacians, and Matching Polynomials of n-Dimensional Hypercubes

We obtain a large number of degree and distance-based topological indices, graph and Laplacian spectra and the corresponding polynomials, entropies and matching polynomials of n-dimensional hypercubes through the use of Hadamard symmetry and recursive dynamic computational techniques. Moreover, computations are used to provide independent numerical values for the topological indices of the 11- and 12-cubes. We invoke symmetry-based recursive Hadamard transforms to obtain the graph and Laplacian spectra of nD-hypercubes and the computed numerical results are constructed for up to 23-dimensional hypercubes. The symmetries of these hypercubes constitute the hyperoctahedral wreath product groups which also pave the way for the symmetry-based elegant computations. These results are used to independently validate the exact analytical expressions that we have obtained for the topological indices as well as graph, Laplacian spectra and their polynomials. We invoke a robust dynamic programming technique to handle the computationally intensive generation of matching polynomials of hypercubes and compute all matching polynomials up to the 6-cube. The distance degree sequence vectors have been obtained numerically for up to 108-dimensional cubes and their frequencies are found to be in binomial distributions akin to the spectra of n-cubes.

The Eulerian distribution on the fixed-point free involutions of the hyperoctahedral group

The Eulerian distribution on the fixed-point free involutions of the hyperoctahedral group

Reflection length with two parameters in the asymptotic representation theory of type B/C and applications

We introduce a two-parameter function ϕq+,q− on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length. We show that this signed reflection function ϕq+,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.

Actions of the hyperoctahedral group to compute minimal contractors

The hyperoctahedral group Bn is the group of symmetries of the hypercube [−1,1]n of Rn. For instance permutations, or symmetries along each of the n canonical planes of Rn all belong to Bn. Now, many sets of equations contain symmetries in Bn. This is the case of the addition constraint: x1+x2=x3 or the multiplication x1⋅x2=x3. In robotics, many specific geometrical constraints such as for instance constraints involving distances or angles used for localization also have these symmetries. This paper shows the fundamental role of the hyperoctahedral group for interval-based methods. These methods use operators, called contractors, which contract axis-aligned boxes, without removing any point of the solution set defined by a conjunction of constraints (typically equations, or inequalities). More precisely, the paper presents an algorithm which allows us to build minimal contractors associated to constraints with symmetries in Bn. As an application, we will consider the geometrical constraint associated to the angle between vectors. The corresponding contractor will then be used in a constraint propagation framework in order to localize a robot using several radars.

Traces on diagram algebras II: centralizer algebras of easy groups and new variations of the Young graph

In continuation of our recent work arXiv:2006.07312, we classify the extremal traces on infinite diagram algebras that appear in the context of Schur-Weyl duality for Banica and Speicher's easy groups. We show that the branching graphs of these algebras describe walks on new variations of the Young graph which describe curious ways of growing Young diagrams. As a consequence, we prove that the extremal traces on generic rook-Brauer algebras are always extensions of extremal traces on the group algebra $\mathbb{C}[S_{\infty}]$ of the infinite symmetric group. Moreover, we conjecture that the same is true for generic parameter deformations of the centralizers of the hyperoctahedral group and we reduce this conjecture to a conceptually much simpler numerical statement. Lastly, we address the trace classification problem for the Schur-Weyl dual of the halfliberated orthogonal group $O_N^*$, in which case extremal traces are always extensions of extremal traces on $\mathbb{C}[S_{\infty} \times S_{\infty}]$. Our approach relies on methods developed by Vershik and Nikitin.

The hull metric on Coxeter groups

We reinterpret an inequality, due originally to Sidorenko, for linear extensions of posets in terms of convex subsets of the symmetric group $\mathfrak{S}_n$. We conjecture that the analogous inequalities hold in arbitrary (not-necessarily-finite) Coxeter groups $W$, and prove this for the hyperoctahedral groups $B_n$ and all right-angled Coxeter groups. Our proof for $B_n$ (and new proof for $\mathfrak{S}_n$) use a combinatorial insertion map closely related to the well-studied promotion operator on linear extensions; this map may be of independent interest. We also note that the inequalities in question can be interpreted as a triangle inequalities, so that convex hulls can be used to define a new invariant metric on $W$ whenever our conjecture holds. Geometric properties of this metric are an interesting direction for future research.

On characters of wreath products

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.Mathematics Subject Classifications: 20C30, 20E22, 05E10Keywords: Character identity, wreath product, partition, Murnaghan-Nakayama rule, colored permutations

Specht property for the algebra of upper triangular matrices of size two with a Taft’s algebra action

AbstractLet F be a field of characteristic zero, and let $UT_2$ be the algebra of $2 \times 2$ upper triangular matrices over F. In a previous paper by Centrone and Yasumura, the authors give a description of the action of Taft’s algebras $H_m$ on $UT_2$ and its $H_m$ -identities. In this paper, we give a complete description of the space of multilinear $H_m$ -identities in the language of Young diagrams through the representation theory of the hyperoctahedral group. We finally prove that the variety of $H_m$ -module algebras generated by $UT_2$ has the Specht property, i.e., every $T^{H_m}$ -ideal containing the $H_m$ -identities of $UT_2$ is finitely based.

Admissible Reversing and Extended Symmetries for Bijective Substitutions

In this paper, we deal with reversing and extended symmetries of subshifts generated by bijective substitutions. We survey some general algebraic and dynamical properties of these subshifts and recall known results regarding their symmetry groups. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended symmetry, and algorithms how to compute them. Moreover, for any finite group H and any subgroup P of the d-dimensional hyperoctahedral group, we construct a bijective substitution which generates an aperiodic subshift with symmetry group {mathbb {Z}}^{d}times H and extended symmetry group ({mathbb {Z}}^{d} rtimes P)times H. A similar construction with the same symmetry group, but with extended symmetry group ({mathbb {Z}}^{d} times H) rtimes P is also provided under a mild assumption on the dimension.

Free quantum analogue of Coxeter group D4

We define the quantum group D4+ – a free quantum version of the demihyperoctahedral group D4 (the smallest representative of the Coxeter series D). In order to do so, we construct a free analogue of the property that a 4×4 matrix has determinant one. Such analogues of determinants are usually very hard to define for free quantum groups in general and our result only holds for the matrix size N=4. The free D4+ is then defined by imposing this generalized determinant condition on the free hyperoctahedral group H4+. Moreover, we give a detailed combinatorial description of the representation category of D4+.

Eulerian pairs and Eulerian recurrence systems

In this paper, we introduce the definitions of Eulerian pair and Hermite-Biehler pair. We also characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems. This generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including up-down run polynomials for symmetric groups, alternating run polynomials for hyperoctahedral groups, flag descent polynomials for hyperoctahedral groups and flag ascent-plateau polynomials for Stirling permutations. We derive some properties of associated polynomials. In particular, we prove the alternatingly increasing property and the interlacing property of the ascent-plateau and left ascent-plateau polynomials for Stirling permutations.

Block number, descents and Schur positivity of fully commutative elements in [formula omitted

The distribution of Coxeter descents and block number over the set of fully commutative elements in the hyperoctahedral group Bn, FC(Bn), is studied in this paper. We prove that the associated Chow quasi-symmetric generating function is equal to a non-negative sum of products of two Schur functions. The proof involves a decomposition of FC(Bn) into a disjoint union of two-sided Barbash–Vogan combinatorial cells, a type B extension of Rubey’s descent preserving involution on 321-avoiding permutations and a detailed study of the intersection of FC(Bn) with Sn-cosets which yields a new decomposition of FC(Bn) into disjoint subsets called fibers. We also compare two different type B Schur-positivity notions, arising from works of Chow and Poirier.