scholarly journals Supercharacters of Unipotent Groups

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Scott Andrews
Keyword(s):  
Unitary Group ◽  
Group Actions ◽  
Large Collection ◽  
Set Partitions ◽  
Group Case ◽  
Unipotent Groups ◽  
International Audience ◽  
Symplectic Matrices ◽  
Algebra Groups

International audience \textbfAbstract. We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters of algebra groups. The resulting supercharacter theories agree with those of André and Neto in the case of the unipotent orthogonal and symplectic matrices and generalize to a large collection of subgroups. In the unitary group case, we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table. \bigbreak

scholarly journals Arc-Coloured Permutations

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Lily Yen
Keyword(s):  
Generating Functions ◽  
Joint Distribution ◽  
Rational Functions ◽  
Set Partitions ◽  
Combinatorial Objects ◽  
Crossing Numbers ◽  
Rational Series ◽  
Nous Montrons ◽  
International Audience ◽  
Labelled Graphs

International audience The equidistribution of many crossing and nesting statistics exists in several combinatorial objects like matchings, set partitions, permutations, and embedded labelled graphs. The involutions switching nesting and crossing numbers for set partitions given by Krattenthaler, also by Chen, Deng, Du, Stanley, and Yan, and for permutations given by Burrill, Mishna, and Post involved passing through tableau-like objects. Recently, Chen and Guo for matchings, and Marberg for set partitions extended the result to coloured arc annotated diagrams. We prove that symmetric joint distribution continues to hold for arc-coloured permutations. As in Marberg's recent work, but through a different interpretation, we also conclude that the ordinary generating functions for all j-noncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions. We use the interpretation to automate the generation of these rational series for both noncrossing and nonnesting coloured set partitions and permutations. <begin>otherlanguage*</begin>french L'équidistribution de plusieurs statistiques décrites en termes d'emboitements et de chevauchements d'arcs s'observes dans plusieurs familles d'objects combinatoires, tels que les couplages, partitions d'ensembles, permutations et graphes étiquetés. L'involution échangeant le nombre d'emboitements et de chevauchements dans les partitions d'ensemble due à Krattenthaler, et aussi Chen, Deng, Du, Stanley et Yan, et l'involution similaire dans les permutations due à Burrill, Mishna et Post, requièrent d'utiliser des objets de type tableaux. Récemment, Chen et Guo pour les couplages, et Marberg pour les partitions d'ensembles, ont étendu ces résultats au cas de diagrammes arc-annotés coloriés. Nous démontrons que la propriété d'équidistribution s'observe est aussi vraie dans le cas de permutations aux arcs coloriés. Tout comme dans le travail résent de Marberg, mais via un autre chemin, nous montrons que les séries génératrices ordinaires des permutations r-coloriées ayant au plus j chevauchements et k emboitements, comptées selon la taille n, sont des fonctions rationnelles. Nous décrivons aussi des algorithmes permettant de calculer ces fonctions rationnelles pour les partitions d'ensembles et les permutations coloriées sans emboitement ou sans chevauchement. <end>otherlanguage*</end>

scholarly journals The Mixing Time for a Random Walk on the Symmetric Group Generated by Random Involutions

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Megan Bernstein
Keyword(s):  
Information Theory ◽  
Random Walk ◽  
Symmetric Group ◽  
Random Walks ◽  
Binomial Distribution ◽  
Unitary Group ◽  
Mixing Time ◽  
Sufficient Time ◽  
International Audience ◽  
A New Technique

International audience The involution walk is a random walk on the symmetric group generated by involutions with a number of 2-cycles sampled from the binomial distribution with parameter p. This is a parallelization of the lazy transposition walk onthesymmetricgroup.Theinvolutionwalkisshowninthispapertomixfor1 ≤p≤1fixed,nsufficientlylarge 2 in between log1/p(n) steps and log2/(1+p)(n) steps. The paper introduces a new technique for finding eigenvalues of random walks on the symmetric group generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. This is especially efficient at calculating the large eigenvalues. The smaller eigenvalues are handled by developing monotonicity relations that also give after sufficient time the likelihood order, the order from most likely to least likely state. The walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of back holes.

Vectors with orthogonal iterates along IP-sets in unitary group actions

2002 ◽  
Vol 22 (01) ◽  
Author(s):  
ANDRÉS DEL JUNCO
Keyword(s):  
Unitary Group ◽  
Group Actions

scholarly journals ON A CERTAIN LOCAL IDENTITY FOR LAPID–MAO'S CONJECTURE AND FORMAL DEGREE CONJECTURE : EVEN UNITARY GROUP CASE

2021 ◽  
pp. 1-55
Author(s):  
Kazuki Morimoto
Keyword(s):  
Explicit Formula ◽  
Unitary Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Unitary Groups ◽  
Reductive Groups ◽  
Local Identity ◽  
Metaplectic Groups ◽  
Group Case ◽  
Formal Degree

Abstract Lapid and Mao formulated a conjecture on an explicit formula of Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups and metaplectic groups as an analogue of the Ichino–Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this article, we study the even unitary-group case. Indeed, we prove this local identity over p-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over p-adic fields and get an explicit formula of Whittaker–Fourier coefficients under certain assumptions.

scholarly journals Tamari Lattices for Parabolic Quotients of the Symmetric Group

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Henri Mühle ◽  
Nathan Williams
Keyword(s):  
Symmetric Group ◽  
Coxeter Groups ◽  
Weak Order ◽  
Set Partitions ◽  
Tamari Lattice ◽  
Nous Montrons ◽  
International Audience ◽  
Tamari Lattices

International audience We present a generalization of the Tamari lattice to parabolic quotients of the symmetric group. More precisely, we generalize the notions of 231-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients, and show bijectively that these sets are equinumerous. Furthermore, the restriction of weak order on the parabolic quotient to the parabolic 231-avoiding permutations is a lattice quotient. Lastly, we suggest how to extend these constructions to all Coxeter groups. Nous présentons une généralisation du treillis de Tamari aux quotients paraboliques du groupe symétrique. Plus précisément, nous généralisons les notions de permutations qui évitent le motif 231, les partitions non-croisées, et les partitions non-emboîtées aux quotients paraboliques, et nous montrons de façon bijective que ces ensembles sont équipotents. En restreignant l’ordre faible du quotient parabolique aux permutations paraboliques qui évitent le motif 231, on obtient un quotient de treillis d’ordre faible. Enfin, nous suggérons comment étendre ces constructions à tous les groupes de Coxeter.

scholarly journals Partition and composition matrices: two matrix analogues of set partitions

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Anders Claesson ◽  
Mark Dukes ◽  
Martina Kubitzke
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Set Partitions ◽  
Nous Montrons ◽  
International Audience ◽  
One To One ◽  
Natural Result ◽  
Ascent Sequences ◽  
Nous Utilisons

International audience This paper introduces two matrix analogues for set partitions; partition and composition matrices. These two analogues are the natural result of lifting the mapping between ascent sequences and integer matrices given in Dukes & Parviainen (2010). We prove that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel. We show that composition matrices on the set $X$ are in one-to-one correspondence with (2+2)-free posets on $X$.We show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on $\{1,\ldots,n\}$. Ce papier introduit deux analogues matriciels des partitions d'ensembles: les matrices de composition et de partition. Ces deux analogues sont le produit naturel du relèvement de l'application entre suites de montées et matrices d'entiers introduite dans Dukes & Parviainen (2010). Nous démontrons que les matrices de partition sont en bijection avec les tables d'inversion, les tables d'inversion croissantes correspondant aux matrices de partition avec une relation d'ordre sur les lignes. Les matrices de partition s-diagonales sont classées en fonction de leurs tables d'inversion. Les matrices de partition bidiagonales sont énumérées par la méthode de matrices de transfert et ont même cardinalité que les permutations triables par deux piles en parallèle. Nous montrons que les matrices de composition sur l'ensemble $X$ sont en bijection avec les ensembles ordonnés (2+2)-libres sur $X$. Nous prouvons que les paires de suites de montées et de permutations sont en bijection avec les ensembles ordonnés (2+2)-libres dont les éléments sont les cycles d'une permutation, et nous utilisons cette relation pour exprimer le nombre d'ensembles ordonnés (2+2)-libres sur $\{1,\ldots,n\}$.

scholarly journals Coloring Rings in Species

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jacob White
Keyword(s):  
Directed Graph ◽  
Symmetric Function ◽  
Chromatic Polynomial ◽  
Connected Components ◽  
Expansion Formula ◽  
Strongly Connected Components ◽  
Set Partitions ◽  
Strongly Connected ◽  
International Audience ◽  
Chromatic Symmetric Function

International audience We present a generalization of the chromatic polynomial, and chromatic symmetric function, arising in the study of combinatorial species. These invariants are defined for modules over lattice rings in species. The primary examples are graphs and set partitions. For these new invariants, we present analogues of results regarding stable partitions, the bond lattice, the deletion-contraction recurrence, and the subset expansion formula. We also present two detailed examples, one related to enumerating subgraphs by their blocks, and a second example related to enumerating subgraphs of a directed graph by their strongly connected components.

scholarly journals Nonzero coefficients in restrictions and tensor products of supercharacters of $U_n(q)$ (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Stephen Lewis ◽  
Nathaniel Thiem
Keyword(s):  
Tensor Product ◽  
Symmetric Group ◽  
Nonnegative Integer ◽  
Set Partitions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Branching Rules ◽  
International Audience ◽  
Supercharacter Theory ◽  
Donne Lieu

International audience The standard supercharacter theory of the finite unipotent upper-triangular matrices $U_n(q)$ gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of $U_m(q) \subseteq U_n(q)$ for $m \leq n$ lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of $U_n(q)$ is a nonnegative integer linear combination of supercharacters of $U_m(q)$ (in fact, it is polynomial in $q$). In a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of $U_n(q)$, this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters. These conditions are given uniformly in terms of complete matchings in bipartite graphs. La théorie standard des supercaractères des matrices triangulaires supérieures unipotentes finies $U_n(q)$ donne lieu à une merveilleuse combinatoire basée sur les partitions d'ensembles. Comme avec la théorie des représentations du groupe symétrique, Les plongements $U_m(q) \subseteq U_n(q)$ pour $m \leq n$ mènent aux règles de branchement. Diaconis et Isaacs ont montré que la restriction d'un supercaractère de $U_n(q)$ est une combinaison linéaire des supercaractères de $U_m(q)$ avec des coefficients entiers non négatifs (en fait, elle est polynomiale en $q$). Dans une première étape vers la compréhension de la combinatoire des coefficients dans les règles de branchement des supercaractères de $U_n(q)$, ce texte caractérise les coefficients non nuls dans la restriction d'un supercaractère et dans le produit des tenseurs de deux supercaractères. Ces conditions sont données uniformément en termes des couplages complets dans des graphes bipartis.

scholarly journals Generating trees for partitions and permutations with no k-nestings

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Sophie Burrill ◽  
Sergi Elizalde ◽  
Marni Mishna ◽  
Lily Yen
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Young Tableaux ◽  
Set Partitions ◽  
Lattice Walks ◽  
Generating Trees ◽  
Generating Tree ◽  
International Audience ◽  
Tree Approach

International audience We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter. Nous décrivons une approche, basée sur l'utilisation d'arbres de génération, pour énumération et la génération exhaustive de partitions et permutations sans k-emboîtement. Contrairement aux travaux antérieurs qui reposent sur un lien entre ces objets, tableaux de Young et familles de chemins dans des treillis, notre approche traite directement partitions et diagrammes de permutations. Nous fournissons des équations fonctionnelles explicites pour les séries génératrices, avec k en tant que paramètre.

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