scholarly journals On Hyperoctahedral Enumeration System, Application to Signed Permutations

2020 ◽  
pp. 40-49
Author(s):  
Iharantsoa Vero Raharinirina
Keyword(s):  
System Application ◽  
Number System ◽  
Lexicographical Order ◽  
Hyperoctahedral Group ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Basic Facts

In this paper, we give the denitions and basic facts about hyperoctahedral number system. There is a natural correspondence between the integers expressed in the latter and the elements of the hyperoctahedral group when we use the inversion statistic on this group to code the signed permutations. We show that this correspondence provides a way with which the signed permutations group can be ordered. With this classication scheme, we can nd the r-th signed permutation from a given number r and vice versa without consulting the list in lexicographical order of the elements of the signed permutations group.

scholarly journals Diagrams and Essential Sets for Signed Permutations

10.37236/8106 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
David Anderson
Keyword(s):  
Schubert Variety ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Degeneracy Locus ◽  
Essential Set ◽  
Diagrammatic Method ◽  
Essential Sets ◽  
Rank Conditions ◽  
Theoretic Version

We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations.  In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation.  Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.

scholarly journals A Basis for the Diagonally Signed-Symmetric Polynomials

10.37236/3224 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
José Manuel Gómez
Keyword(s):  
Polynomial Ring ◽  
Symmetric Polynomials ◽  
Hyperoctahedral Group ◽  
Free Basis ◽  
Signed Permutations ◽  
Invariant Ring

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.

scholarly journals Generating All 36,864 Four-Color Adinkras via Signed Permutations and Organizing into ℓ- and ℓ˜-Equivalence Classes

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 120 ◽  
Author(s):  
S. Gates ◽  
Kevin Iga ◽  
Lucas Kang ◽  
Vadim Korotkikh ◽  
Kory Stiffler
Keyword(s):  
Permutation Group ◽  
Quaternion Algebra ◽  
Equivalence Classes ◽  
Inner Product ◽  
Complete Analysis ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Non Linear

Recently, all 1,358,954,496 values of the gadget between the 36,864 adinkras with four colors, four bosons, and four fermions have been computed. In this paper, we further analyze these results in terms of B C 3 , the signed permutation group of three elements, and B C 4 , the signed permutation group of four elements. It is shown how all 36,864 adinkras can be generated via B C 4 boson × B C 3 color transformations of two quaternion adinkras that satisfy the quaternion algebra. An adinkra inner product has been used for some time, known as the gadget, which is used to distinguish adinkras. We show how 96 equivalence classes of adinkras that are based on the gadget emerge in terms of B C 3 and B C 4 . We also comment on the importance of the gadget as it relates to separating out dynamics in terms of Kähler-like potentials. Thus, on the basis of the complete analysis of the supersymmetrical representations achieved in the preparatory first four sections, the final comprehensive achievement of this work is the construction of the universal B C 4 non-linear σ -model.

scholarly journals Depth in Coxeter groups of type $B$

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Eli Bagno ◽  
Riccardo Biagioli ◽  
Mordechai Novick
Keyword(s):  
Coxeter Group ◽  
Simple Formula ◽  
Coxeter Groups ◽  
Type B ◽  
Minimal Path ◽  
Signed Permutation ◽  
Signed Permutations ◽  
International Audience ◽  
Path Cost

International audience The depth statistic was defined for every Coxeter group in terms of factorizations of its elements into product of reflections. Essentially, the depth gives the minimal path cost in the Bruaht graph, where the edges have prescribed weights. We present an algorithm for calculating the depth of a signed permutation which yields a simple formula for this statistic. We use our algorithm to characterize signed permutations having depth equal to length. These are the fully commutative top-and-bottom elements defined by Stembridge. We finally give a characterization of the signed permutations in which the reflection length coincides with both the depth and the length. La statistique profondeur a été introduite par Petersen et Tenner pour tout groupe de Coxeter $W$. Elle est définie pour tout $w \in W$ à partir de ses factorisations en produit de réflexions (non nécessairement simples). Pour le type $B$, nous introduisons un algorithme calculant la profondeur, et donnant une formule explicite pour cette statistique. On utilise par ailleurs cet algorithme pour caractériser tous les éléments ayant une profondeur égale à leur longueur. Ces derniers s’avèrent être les éléments pleinement commutatifs “hauts-et-bas” introduits par Stembridge. Nous donnons enfin une caractérisation des éléments dont la longueur absolue, la profondeur et la longueur coïncident.

scholarly journals MacMahon-type Identities for Signed Even Permutations

10.37236/1836 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Dan Bernstein
Keyword(s):  
Symmetric Group ◽  
Alternating Group ◽  
Hyperoctahedral Group ◽  
Major Index ◽  
Signed Permutations ◽  
Natural Analogues ◽  
Classic Theorem

MacMahon's classic theorem states that the length and major index statistics are equidistributed on the symmetric group $S_n$. By defining natural analogues or generalizations of those statistics, similar equidistribution results have been obtained for the alternating group $A_n$ by Regev and Roichman, for the hyperoctahedral group $B_n$ by Adin, Brenti and Roichman, and for the group of even-signed permutations $D_n$ by Biagioli. We prove analogues of MacMahon's equidistribution theorem for the group of signed even permutations and for its subgroup of even-signed even permutations.

scholarly journals Theta-Vexillary Signed Permutations

10.37236/8023 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Jordan Lambert
Keyword(s):  
Pattern Avoidance ◽  
Type B ◽  
Hyperoctahedral Group ◽  
Degeneracy Loci ◽  
Signed Permutations

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.

scholarly journals Asymptotics of a Locally Dependent Statistic on Finite Reflection Groups

10.37236/9454 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Frank Röttger
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Coxeter Groups ◽  
Random Permutation ◽  
Wasserstein Distance ◽  
Reflection Groups ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Dependency Structures

This paper discusses the asymptotic behaviour of the number of descents in a random signed permutation and its inverse, which was listed as an interesting direction by Chatterjee and Diaconis (2017). 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 to signed permutations, i.e. elements of Coxeter groups of type $ \mathtt{B}_n $, which are also known as the hyperoctahedral groups.  Furthermore, a similar central limit theorem for elements of Coxeter groups of type $\mathtt{D}_n$ 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 Cramér-Wold device.

scholarly journals Words and Noncommutative Invariants of the Hyperoctahedral Group

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Anouk Bergeron-Brlek
Keyword(s):  
Tensor Algebra ◽  
Combinatorial Method ◽  
Complex Vector ◽  
Hyperoctahedral Group ◽  
Signed Permutation ◽  
Combinatorial Interpretations ◽  
Free Generators ◽  
International Audience ◽  
Espace Vectoriel ◽  
Nous Utilisons

International audience Let $\mathcal{B}_n$ be the hyperoctahedral group acting on a complex vector space $\mathcal{V}$. We present a combinatorial method to decompose the tensor algebra $T(\mathcal{V})$ on $\mathcal{V}$ into simple modules via certain words in a particular Cayley graph of $\mathcal{B}_n$. We then give combinatorial interpretations for the graded dimension and the number of free generators of the subalgebra $T(\mathcal{V})^{\mathcal{B}_n}$ of invariants of $\mathcal{B}_n$, in terms of these words, and make explicit the case of the signed permutation module. To this end, we require a morphism from the Mantaci-Reutenauer algebra onto the algebra of characters due to Bonnafé and Hohlweg. Soit $\mathcal{B}_n$ le groupe hyperoctaédral agissant sur un espace vectoriel complexe $\mathcal{V}$. Nous présentons une méthode combinatoire donnant la décomposition de l'algèbre $T(\mathcal{V})$ des tenseurs sur $\mathcal{V}$ en modules simples via certains mots dans un graphe de Cayley donné. Nous donnons ensuite des interprétations combinatoires pour la dimension graduée et le nombre de générateurs libres de la sous-algèbre $T(\mathcal{V})^{\mathcal{B}_n}$ des invariants de $\mathcal{B}_n$, en termes de ces mots, et explicitons le cas du module de permutation signé. À cette fin, nous utilisons un morphisme entre l'algèbre de Mantaci-Reutenauer et l'algèbre des caractères introduit par Bonnafé et Hohlweg.

TWO NOTES ON GENOME REARRANGEMENT

2003 ◽  
Vol 01 (01) ◽  
pp. 71-94 ◽  
Author(s):  
MICHAL OZERY-FLATO ◽  
RON SHAMIR
Keyword(s):  
Lower Bound ◽  
Genome Rearrangement ◽  
Polynomial Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Minimum Number ◽  
The Common ◽  
Signed Genome

A central problem in genome rearrangement is finding a most parsimonious rearrangement scenario using certain rearrangement operations. An important problem of this type is sorting a signed genome by reversals and translocations (SBRT). Hannenhalli and Pevzner presented a duality theorem for SBRT which leads to a polynomial time algorithm for sorting a multi-chromosomal genome using a minimum number of reversals and translocations. However, there is one case for which their theorem and algorithm fail. We describe that case and suggest a correction to the theorem and the polynomial algorithm. The solution of SBRT uses a reduction to the problem of sorting a signed permutation by reversals (SBR). The best extant algorithms for SBR require quadratic time. The common approach to solve SBR is by finding a safe reversal using the overlap graph or the interleaving graph of a permutation. We describe a family of signed permutations which proves a quadratic lower bound on the number of affected vertices in the overlap/interleaving graph during any optimal sorting scenario. This implies, in particular, an Ω(n3) lower bound for Bergeron's algorithm.

Using Patterns to Practice Basic Facts

1990 ◽  
Vol 37 (8) ◽  
pp. 38-41
Author(s):  
Miriam M. Feinberg
Keyword(s):  
Number System ◽  
Use Patterns ◽  
Finger Counting ◽  
Mathematical Problems ◽  
Addition And Subtraction ◽  
Insurmountable Problem ◽  
Basic Facts ◽  
Subtraction Facts

Memorizing the basic addition and subtraction facts becomes an insurmountable problem for many pupils, and finger counting remains their basic counting tool in the middle and upper grades. However, if they recognize and use patterns, they can develop a better understanding of the number system. An understanding of patterns then becomes an important tool as they progress to increasingly complex mathematical problems.

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