scholarly journals The Sorting Order on a Coxeter Group

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Drew Armstrong

International audience Let $(W,S)$ be an arbitrary Coxeter system. For each sequence $\omega =(\omega_1,\omega_2,\ldots) \in S^{\ast}$ in the generators we define a partial order― called the $\omega \mathsf{-sorting order}$ ―on the set of group elements $W_{\omega} \subseteq W$ that occur as finite subwords of $\omega$ . We show that the $\omega$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and strong Bruhat orders on the group. Moreover, the $\omega$-sorting order is a "maximal lattice'' in the sense that the addition of any collection of edges from the Bruhat order results in a nonlattice. Along the way we define a class of structures called $\mathsf{supersolvable}$ $\mathsf{antimatroids}$ and we show that these are equivalent to the class of supersolvable join-distributive lattices.

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Bridget Eileen Tenner

International audience The Bruhat order gives a poset structure to any Coxeter group. The ideal of elements in this poset having boolean principal order ideals forms a simplicial poset. This simplicial poset defines the boolean complex for the group. In a Coxeter system of rank n, we show that the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of these spheres is the boolean number, which can be computed inductively from the unlabeled Coxeter system, thus defining a graph invariant. For certain families of graphs, the boolean numbers have intriguing combinatorial properties. This work involves joint efforts with Claesson, Kitaev, and Ragnarsson. \par L'ordre de Bruhat munit tout groupe de Coxeter d'une structure de poset. L'idéal composé des éléments de ce poset engendrant des idéaux principaux ordonnés booléens, forme un poset simplicial. Ce poset simplicial définit le complexe booléen pour le groupe. Dans un système de Coxeter de rang n, nous montrons que le complexe booléen est homotopiquement équivalent à un bouquet de sphères de dimension (n-1). Le nombre de ces sphères est le nombre booléen, qui peut être calculé inductivement à partir du système de Coxeter non-étiquetté; définissant ainsi un invariant de graphe. Pour certaines familles de graphes, les nombres booléens satisfont des propriétés combinatoires intriguantes. Ce travail est une collaboration entre Claesson, Kitaev, et Ragnarsson.


2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Soojin Cho ◽  
Kyoungsuk Park

International audience Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. We find the relation between the number of alignments, crossings and other statistics of signed permutations, and also characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$. De nombreuses statistiques importantes des permutations signées sont réalisées dans les tableaux de permutations ou ”bare” tableaux de type $B$ correspondants : les alignements, croisements et inversions des permutations signées sont réalisés dans les tableaux de permutations de type $B$ correspondants, et les cycles des permutations signées sont comprises dans les ”bare” tableaux de type $B$ correspondants. Cela nous mène à relier le nombre d’alignements et de croisements avec d’autres statistiques des permutations signées, et aussi de caractériser la relation de couverture dans l’ordre de Bruhat faible sur des systèmes de Coxeter de type $B$ en termes de tableaux de permutations de type $B$.


10.37236/4942 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Henri Mühle

In this article, we investigate the set of $\gamma$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $\gamma\in W$, under Bruhat order, and we denote this poset by $\mathcal{B}_{\gamma}$. We show that this poset belongs to the class of SB-lattices recently introduced by Hersh and Mészáros, by proving a more general statement, namely that all join-distributive lattices are SB-lattices. The observation that $\mathcal{B}_{\gamma}$ is join-distributive is due to Armstrong. Subsequently, we investigate for which finite Coxeter groups $W$ and which Coxeter elements $\gamma\in W$ the lattice $\mathcal{B}_{\gamma}$ is in fact distributive. It turns out that this is the case for the "coincidental" Coxeter groups, namely the groups $A_{n},B_{n},H_{3}$ and $I_{2}(k)$. We conclude this article with a conjectural characteriziation of the Coxeter elements $\gamma$ of said groups for which $\mathcal{B}_{\gamma}$ is distributive in terms of forbidden orientations of the Coxeter diagram.


Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss

This chapter introduces the notion of a Tits index and the notion of the relative Coxeter diagram of a Tits index. It first defines a Tits index, which can be anisotropic or isotropic, quasi-split or split, before considering a number of propositions regarding compatible representations. It then gives a proof of the theorem that includes two assumptions about a Coxeter system, focusing on the absolute Coxeter system, the relative Coxeter system, and the relative Coxeter group of the Tits index, as well as the absolute Coxeter diagram (or absolute type), the relative Coxeter diagram (or relative type), and the absolute rank and the relative rank of the Tits index. The chapter concludes with some observations about the case that (W, S) is spherical, irreducible or affine.


Author(s):  
Gezahagne Mulat Addis

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.


2021 ◽  
Vol 14 (3) ◽  
pp. 207-217
Author(s):  
Tilahun Mekonnen Munie

In the field of many valued logic, lattice valued logic (especially ideals) plays an important role. Nowadays, lattice valued logic is becoming a research area. Researchers introduced weak LI-ideals of lattice implication algebra. Furthermore, other scholars researched LI-ideals of implicative almost distributive lattice. Therefore, the target of this paper was to investigate new development on the extension of LI-ideal theories and properties in implicative almost distributive lattice. So, in this paper, the notion of weak LI-ideals and maximal weak LI- ideals of implicative almost distributive lattice are defined. The properties of weak LI- ideals in implicative almost distributive lattice are studied and several characterizations of weak LI-ideals are given. Relationship between weak LI-ideals and weak filters are explored. Hence, the extension properties of weak LI-ideal of lattice implication algebra to that of weak LI-ideal of implicative almost distributive lattice were shown.


2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Matthieu Latapy

International audience In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.


2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Christian Stump ◽  
Hugh Thomas ◽  
Nathan Williams

International audience The main objects of noncrossing Catalan combinatorics associated to a finite Coxeter system are noncross- ing partitions, sortable elements, and cluster complexes. The first and the third of these have known Fuss–Catalan generalizations. We provide new viewpoints for these, introduce a corresponding generalization of sortable elements as elements in the positive Artin monoid, and show how this perspective ties together all three generalizations.


Slovo ◽  
2017 ◽  
Vol The autobiographical... (Beyond the steppes of Central...) ◽  
Author(s):  
Adeeb KHALID

International audience This a rticle explores t he s o-called “ Memoirs” o f M unavvar qori Abdurashidxon o’g’li (1878-1931), a major figure in the politics of Turkestanin the era of the revolution and an early victim of the OGPU. The autobiographical text is a series of pokazanija written while Munavvar qori was under arrest in which he describes his political activities since the revolution. The article analyses the way in which Munavvar qori presents himself—a combination limited confession with plausible denial or extenuation—and the way he deploys language. The article also presents lengthy excerpts in English translation. cette étude examine les prétendus « Mémoires » de Munavvar qori Abdurashidxon o’g’li (1878-1931), une figure d’importance dans l’histoirepolitique du Turkestan à l’ère révolutionnaire et une des premières victimes de l’OGPU. Le texte autobiographique comprend une série de pokazanija, écrits quand Munavvar qori était en état d’arrestation, et qui racontent ses activités depuis la révolution. L’étude analyse la manière avec laquelle on se présente – soit une combinaison de confession limitée et de démentis vraisemblables – et le langage qu’on y déploie. L’étude offre aussi des extraits du texte en traduction anglaise. В статье исследуется так называемые «Воспоминания» Мунаввар кори Абдурашидхонов (1878-1931 гг.), один из виднейших деятелейполитической жизни Туркестана в эпохи революции и один из ранних жертв ОГПУ. Автобиографический текст состоит из ряда показаний, написанные во время выключения автора, и в которых он запишет его деятельность в послереволюционной период. В данной статье анализируется способов, в которых Мунаввар кори представляет себя (между умеренным признанием вины и отдалением в вероятной мере из ее) и языка, котором он используется. Статья тоже содержат в себе длинных выдержки из текста на английском переводе.


Slovo ◽  
2020 ◽  
Vol How to think of literary... (Varia) ◽  
Author(s):  
Tomasz Krupa

International audience The case of Sorana Gurian (1913‑1956) allows to examine the situation of multiple exclusion in the 20th century European society: she is a Jewish woman, a stranger accused of espionage and collaboration and her body, disabled and affected by cancer, becomes the main culprit of this plural banishment. The author is a tragic figure: not only because of her life, but also due to the oblivion that her oeuvre fell into – yet original and contemporarily recognized – counting six volumes in French and Romanian and dozens of press publications. In the paper, I propose to read chosen literary chronicles commenting on two Gurian’s editions published in Romania in 1945‑1946 – Zilele nu se întorc niciodată [Days that never return] and Întâmplări între amurg și noapte [Adventures between twilight and night]. These chronicles show the way in which the Other was perceived, in this case – a woman writer, a disabled woman or a Jewish woman in the 20th century. Such perspective aims to show the non‑ aesthetic conditions (gender, corporality, social class, political convictions, ethnic origin, etc.) which have determined Gurian’s appearance and disappearance in the literary scene, and which still influence the way of perceiving her texts in Romania. At the end of this study, I reflect on the possibility of rehabilitating this figure in the history of European literature, that could renew the Romanian literary canon of the 20th century, in which women writers haven’t still found their place. Le cas de Sorana Gurian (1913-1956) permet d’examiner la situation d’une multiple exclusion au sein de la société européenne du xxe siècle : elle est à la fois femme, juive et étrangère, accusée d’espionnage et de collaboration, et son corps, handicapé et touché par le cancer, devient le principal coupable de ce bannissement pluriel. L’auteure est une figure tragique : non seulement par sa vie, mais aussi par l’oubli dans lequel est tombée son œuvre – pourtant originale et reconnue à l’époque –, comptant six volumes en roumain et en français et des dizaines de publications dans la presse. Dans le présent article, je propose une lecture de quelques chroniques littéraires portant sur deux éditions de Gurian parues en Roumanie en 1945-1946 : Zilele nu se întorc niciodată [Les jours ne reviennent jamais] et Întâmplări între amurg și noapte [Aventures entre crépuscule et nuit], qui témoignent de la manière dont on percevait l’Autre, en l’occurrence une femme écrivain, une femme infirme ou bien une femme d’origine juive au xxe siècle. Cette perspective a pour but de montrer les conditionnements autres qu’esthétiques (genre, corporalité, classe sociale, convictions politiques, origine ethnique, etc.) qui ont déterminé l’apparition et la disparition de Gurian sur la scène littéraire, et qui influencent toujours la manière de percevoir ses textes en Roumanie. À la fin de cette étude, je réfléchis à la possibilité de réhabiliter cette figure dans l’histoire de la littérature européenne, ce qui pourrait permettre de rediscuter le canon littéraire roumain du xxe siècle, où les femmes écrivains ne trouvent toujours pas leur place. Przypadek Sorany Gurian (1913‑1956) pozwala zbadać sytuację wielokrotnego wykluczenia w xx‑wiecznym społeczeństwie europejskim: jest ona jednocześnie kobietą, Żydówką, obcą, oskarżoną o szpiegostwo i kolaborację, zaś jej ciało, niepełnosprawne i chore na raka, staje się głównym winowajcą tego mnogiego wygnania. Autorka jest postacią tragiczną nie tylko ze względu na jej życie, lecz również ze względu na zapomnienie, w jakie popadła jej oryginalna i uznana przez współczesnych twórczość, na którą składają się sześć wydań w językach francuskim i rumuńskim oraz dziesiątki publikacji w prasie. W niniejszym artykule odczytuję wybrane kroniki literackie komentujące dwa teksty opublikowane przez Gurian w Rumunii w latach 1945‑1946 – Zilele nu se întorc niciodată [Dni nigdy nie powracają] oraz Întâmplări între amurg și noapte [Zdarzenia między zmierzchem a nocą]. Kroniki te pokazują bowiem, w jaki sposób postrzegano Innego, w tym przypadku – pisarkę, niepełnosprawną kobietę czy Żydówkę w xx wieku. Ta perspektywa ma na celu wskazać uwarunkowania nie‑ estetyczne (płeć, cielesność, klasa społeczna, poglądy polityczne, pochodzenie etniczne itd.), które zdecydowały o pojawieniu i zniknięciu Gurian na scenie literackiej, a które ciągle określają postrzeganie jej tekstów w Rumunii. Na końcu tego studium zastanawiam się nad możliwością rehabilitacji tej postaci w historii literatury europejskiej, co mogłoby z kolei przyczynić się do odnowienia rumuńskiego kanonu literackiego xx wieku, w którym nadal nie ma miejsca dla pisarek. Cazul Soranei Gurian (1913‑1956) ilustrează situația unei excluderi din considerențe multiple în societatea europeană din secolul xx‑lea. Fiind femeie, evreică și străină, acuzată de spionaj și de colaborare, trupul său, cu handicap și atins de cancer, devine principalul vinovat al acestei exilări plurale. Autorea este un personaj tragic: nu numai din cauza vieții sale, ci și din cauza uitării în care a căzut opera sa, totuși originală și recunoscută de către contemporanii ei, compusă din șase volume în limba franceză și în română, precum și de zeci de publicații în presă. În acest articol, propun o lectură a unor cronici literare privind cele două ediții ale Soranei Gurian publicate în România în anii 1945‑1946 – Zilele nu se întorc niciodată și Aventuri între amurg și noapte. Aceste cronici reflectă modul în care se percepe Celălalt, în acest caz – o scriitoare, o femeie cu handicap, o evreică în contextul secolului xx. Această perspectivă are să identifice diferite condiționări (gen, corporalitate, clasă socială, convingeri politice, etnie șamd), care au determinat atît apariția, cît și dispariția lui Gurian de pe scenă literară și care încă înfluențează modul de percepere al textelor sale în România. La sfîrșitul acestui studiu, propun o reflecție asupra posibilității de reabilitare a acestei figuri în istoria literaturii europene, ceea ce ar putea conduce la rediscutarea canonului literar românesc al secolului xx, unde scriitoarele încă nu‑și găsesc locul.


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