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 Rough sets in graphs using similarity relations

2022 ◽  
Vol 7 (4) ◽  
pp. 5790-5807
Author(s):  
Imran Javaid ◽  
◽  
Shahroz Ali ◽  
Shahid Ur Rehman ◽  
Aqsa Shah
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Sufficient Conditions ◽  
Membership Functions ◽  
Similarity Relations ◽  
Complete Bipartite Graphs ◽  
Essential Set ◽  
Essential Sets ◽  
Necessary And Sufficient ◽  
Clustering Criterion

<abstract><p>In this paper, we investigate the theory of rough set to study graphs using the concept of orbits. Rough sets are based on a clustering criterion and we use the idea of similarity of vertices under automorphism as a criterion. We introduce indiscernibility relation in terms of orbits and prove necessary and sufficient conditions under which the indiscernibility partitions remain the same when associated with different attribute sets. We show that automorphisms of the graph $ \mathcal{G} $ preserve the indiscernibility partitions. Further, we prove that for any graph $ \mathcal{G} $ with $ k $ orbits, any reduct $ \mathcal{R} $ consists of one element from $ k-1 $ orbits of the graph. We also study the rough membership functions for paths, cycles, complete and complete bipartite graphs. Moreover, we introduce essential sets and discernibility matrices induced by orbits of graphs and study their relationship. We also prove that every essential set consists of union of any two orbits of the graph.</p></abstract>

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 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.

IRREDUCIBLE DECOMPOSITIONS OF DEGENERACY LOCI OF MATRICES

2008 ◽  
Vol 18 (02) ◽  
pp. 257-270
Author(s):  
XU-AN ZHAO ◽  
HONGZHU GAO
Keyword(s):  
Degeneracy Loci ◽  
Irreducible Components ◽  
Degeneracy Locus ◽  
Rank Conditions ◽  
Rank Functions

In this paper, we study the irreducible decompositions of the degeneracy loci of matrices which are defined by rank conditions on upper left submatrices. By introducing the concepts of standard and essential rank functions, we give an explicit classification of these degeneracy loci. Based on standard rank functions, we design an algorithm to write a degeneracy locus as a union of its irreducible components. This gives an answer to a problem raised by Sturmfels.

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.

scholarly journals Bijection Between Bigrassmannian Permutations Maximal below a Permutation and its Essential Set

10.37236/476 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Masato Kobayashi
Keyword(s):  
Bruhat Order ◽  
Essential Set ◽  
Essential Sets

Bigrassmannian permutations are known as permutations which have precisely one left descent and one right descent. They play an important role in the study of Bruhat order. Fulton introduced the essential set of a permutation and studied its combinatorics. As a consequence of his work, it turns out that the essential set of bigrassmannian permutations consists of precisely one element. In this article, we generalize this observation for essential sets of arbitrary permutations. Our main theorem says that there exists a bijection between bigrassmanian permutations maximal below a permutation and its essential set. For the proof, we make use of two equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.

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 Polynomial identities related to special Schubert varieties

2021 ◽  
Author(s):  
Francesca Cioffi ◽  
Davide Franco ◽  
Carmine Sessa
Keyword(s):  
Arbitrary Number ◽  
Schubert Variety ◽  
Decomposition Theorem ◽  
Point Of View ◽  
Explicit Description ◽  
Intersection Cohomology ◽  
Schubert Varieties ◽  
Polynomial Identities ◽  
Poincaré Polynomial

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

scholarly journals CFT unitarity and the AdS Cutkosky rules

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
David Meltzer ◽  
Allic Sivaramakrishnan
Keyword(s):  
Flat Space ◽  
Tree Level ◽  
Optical Theorem ◽  
Conformal Field Theories ◽  
Strongly Coupled ◽  
Conformal Field ◽  
Weakly Coupled ◽  
S Matrix ◽  
Space Limit ◽  
Diagrammatic Method

Abstract We derive the Cutkosky rules for conformal field theories (CFTs) at weak and strong coupling. These rules give a simple, diagrammatic method to compute the double-commutator that appears in the Lorentzian inversion formula. We first revisit weakly-coupled CFTs in flat space, where the cuts are performed on Feynman diagrams. We then generalize these rules to strongly-coupled holographic CFTs, where the cuts are performed on the Witten diagrams of the dual theory. In both cases, Cutkosky rules factorize loop diagrams into on-shell sub-diagrams and generalize the standard S-matrix cutting rules. These rules are naturally formulated and derived in Lorentzian momentum space, where the double-commutator is manifestly related to the CFT optical theorem. Finally, we study the AdS cutting rules in explicit examples at tree level and one loop. In these examples, we confirm that the rules are consistent with the OPE limit and that we recover the S-matrix optical theorem in the flat space limit. The AdS cutting rules and the CFT dispersion formula together form a holographic unitarity method to reconstruct Witten diagrams from their cuts.

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