scholarly journals Phylogenetic trees and the tropical geometry of flag varieties

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Christopher Manon
Keyword(s):  
Phylogenetic Trees ◽  
Tropical Geometry ◽  
Type A ◽  
Hyperplane Arrangements ◽  
Nous Allons ◽  
Flag Varieties ◽  
Dynkin Diagrams ◽  
The Family ◽  
International Audience ◽  
Tropical Varieties

International audience We will discuss some recent theorems relating the space of weighted phylogenetic trees to the tropical varieties of each flag variety of type A. We will also discuss the tropicalizations of the functions corresponding to semi-standard tableaux, in particular we relate them to familiar functions from phylogenetics. We close with some remarks on the generalization of these results to the tropical geometry of arbitrary flag varieties. This involves the family of Bergman complexes derived from the hyperplane arrangements associated to simple Dynkin diagrams. Nous allons discuter de quelques théorèmes récents concernant l'espace des arbres phylogénétiques aux variétés Tropicales de chaque variété de drapeaux de type A. Nous allons également discuter des tropicalisations des fonctions correspondant à tableaux semi-standard, en particulier, nous les rapporter à des fonctions familières de la phylogénétique. Nous terminerons avec quelques remarques sur la généralisation de ces résultats à la géométrie tropicale de variétés de drapeaux arbitraires. Il s'agit de la famille de complexes Bergman provenant des arrangements d'hyperplans associés à des diagrammes de Dynkin simples.

scholarly journals Dissimilarity Maps on Trees and the Representation Theory of $GL_n(\mathbb{C})$

10.37236/2715 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Christopher Manon
Keyword(s):  
Representation Theory ◽  
Phylogenetic Trees ◽  
Type A ◽  
Flag Variety ◽  
Flag Varieties ◽  
Tropical Varieties

We revisit representation theory in type $A,$ used previously to establish that the dissimilarity vectors of phylogenetic trees are points on the tropical Grassmannian variety. We use a different version of this construction to show that the space of phylogenetic trees $K_n$ maps to the tropical varieties of every flag variety of $GL_n(\mathbb{C}).$ Using this map, we find a tropical function on the space of phylogenetic trees for each semistandard tableaux, and we show that the functions satisfy the tropicalized equations which cut out $GL_n(\mathbb{C})$ flag varieties.

scholarly journals Hyperplane Arrangements and Diagonal Harmonics

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Drew Armstrong
Keyword(s):  
Symmetric Functions ◽  
Hilbert Series ◽  
Hyperplane Arrangement ◽  
Type A ◽  
Hyperplane Arrangements ◽  
Combinatorial Interpretation ◽  
Nabla Operator ◽  
Affine Weyl Group ◽  
Diagonal Harmonics ◽  
International Audience

International audience In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q,t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement — which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended'' Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions. En 2003, la statistique bounce de Haglund a donné la première interprétation combinatoire de la somme des nombres q,t-Catalan et de la série de Hilbert des harmoniques diagonaux. Dans cet article nous proposons une nouvelle interprétation combinatoire à partir du groupe de Weyl affine de type A. En particulier, nous définissons deux statistiques sur les permutations affines; l'une à partir de l'arrangement d'hyperplans Shi, et l'autre à partir d'un nouvel arrangement — que nous appelons l'arrangement Ish. Nous prouvons que nos statistiques sont équivalentes aux statistiques area' et bounce de Haglund et Loehr. Dans ce contexte, nous observons que bounce s'exprime naturellement comme une statistique sur le réseau des racines. Nous prolongeons nos statistiques dans deux directions: arrangements Shi "étendus'', et chambres bornées associées. Cela conduit à une interprétation (conjecturale) combinatoire pour toutes les puissances entières de l'opérateur nabla de Bergeron-Garsia appliqué aux fonctions symétriques élémentaires.

scholarly journals Chevalley-Monk and Giambelli formulas for Peterson Varieties

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Elizabeth Drellich
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Type A ◽  
Flag Variety ◽  
Flag Varieties ◽  
Dimensional Torus ◽  
One Dimensional ◽  
International Audience ◽  
Schubert Classes ◽  
Partial Flag Varieties

International audience A Peterson variety is a subvariety of the flag variety $G/B$ defined by certain linear conditions. Peterson varieties appear in the construction of the quantum cohomology of partial flag varieties and in applications to the Toda flows. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type A Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives a Chevalley-Monk rule and Giambelli's formula for all Lie types.

scholarly journals Flag Gromov-Witten invariants via crystals

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jennifer Morse ◽  
Anne Schilling
Keyword(s):  
Weyl Group ◽  
Type A ◽  
Schubert Calculus ◽  
Schur Function ◽  
Flag Varieties ◽  
Highest Weight ◽  
Affine Weyl Group ◽  
Nous Montrons ◽  
International Audience ◽  
Complete Flag

International audience We apply ideas from crystal theory to affine Schubert calculus and flag Gromov-Witten invariants. By defining operators on certain decompositions of elements in the type-$A$ affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applied to study the product of a Schur function with a $k$-Schur function. Consequently, we prove that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb{C}^n$ enumerate the highest weight elements under these operators. Nous appliquons des idées provenant de la théorie des bases cristallines au calcul de Schubert affine et aux invariants de drapeaux de Gromov–Witten. Nous définissons des opérateurs sur certaines décompositions d’éléments de groupes de Weyl affines en type $A$ afin de construire une base cristalline encodant la structure interne des modules de Specht associés aux diagrammes de permutations. Nous montrons comment la structure de cristal permet d’étudier le produit d’une fonction de Schur avec une $k$-fonction de Schur. En conséquence, nous prouvons que la sous-classe des invariants de 3-points de Gromov–Witten d’une variété complète de drapeaux complets pour $\mathbb{C}^n$ énumère les éléments de poids maximaux pour ces opérateurs.

scholarly journals Bases for modules of differential operators

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Norihiro Nakashima
Keyword(s):  
Differential Operators ◽  
Type A ◽  
Hyperplane Arrangements ◽  
3 Dimensional ◽  
Modules Of Differential Operators ◽  
International Audience

International audience It is well-known that the derivation modules of Coxeter arrangements are free. Holm began to study the freeness of modules of differential operators on hyperplane arrangements. In this paper, we study the cases of the Coxter arrangements of type A, B and D. In this case, we prove that the modules of differential operators of order 2 are free. We give examples of all the 3-dimensional classical Coxeter arrangements. Two keys for the proof are ``Cauchy–Sylvester's theorem on compound determinants'' and ``Saito–Holm's criterion''. Il est connu que les modules de la dérivation d'arrangements de Coxeter sont libres. Holm a commencè à étudier les modules libres des opérateurs différentiels sur des compositions d'hyperplans. Dans cet article, nous étudions les cas des compositions de Coxter les types A, B et D. Dans ce cas, nous prouvons que les modules d’opérateurs différentiels d'ordre 2 sont libres. Nous donnons des exemples de toutes les compositions de Coxeter classiques de dimension 3. Les deux points clefs pour la preuve sont le théorème de Cauchy–Sylvester sur déterminants composés et le critère de Saito–Holm.

scholarly journals Extensive Phylogenetic Analysis of Piscine Orthoreovirus Genomic Sequences Shows the Robustness of Subgenotype Classification

Pathogens ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 41
Author(s):  
Marcos Godoy ◽  
Daniel A. Medina ◽  
Rudy Suarez ◽  
Sandro Valenzuela ◽  
Jaime Romero ◽  
...  
Keyword(s):  
Phylogenetic Analysis ◽  
Genetic Variation ◽  
Sequence Analysis ◽  
Molecular Phylogeny ◽  
Phylogenetic Trees ◽  
Genomic Segment ◽  
Double Stranded Rna ◽  
The Family ◽  
Original Classification

Piscine orthoreovirus (PRV) belongs to the family Reoviridae and has been described mainly in association with salmonid infections. The genome of PRV consists of about 23,600 bp, with 10 segments of double-stranded RNA, classified as small (S1 to S4), medium (M1, M2 and M3) and large (L1, L2 and L3); these range approximately from 1000 bp (segment S4) to 4000 bp (segment L1). How the genetic variation among PRV strains affects the virulence for salmonids is still poorly understood. The aim of this study was to describe the molecular phylogeny of PRV based on an extensive sequence analysis of the S1 and M2 segments of PRV available in the GenBank database to date (May 2020). The analysis was extended to include new PRV sequences for S1 and M2 segments. In addition, subgenotype classifications were assigned to previously published unclassified sequences. It was concluded that the phylogenetic trees are consistent with the original classification using the PRV genomic segment S1, which differentiates PRV into two major genotypes, I and II, and each of these into two subgenotypes, designated as Ia and Ib, and IIa and IIb, respectively. Moreover, some clusters of country- and host-specific PRV subgenotypes were observed in the subset of sequences used. This work strengthens the subgenotype classification of PRV based on the S1 segment and can be used to enhance research on the virulence of PRV.

scholarly journals Tropical geometry and the motivic nearby fiber

2011 ◽  
Vol 148 (1) ◽  
pp. 269-294 ◽  
Author(s):  
Eric Katz ◽  
Alan Stapledon
Keyword(s):  
Euler Characteristic ◽  
Hodge Structure ◽  
Tropical Geometry ◽  
Mixed Hodge Structure ◽  
General Fiber ◽  
Algebraic Torus ◽  
Tropical Varieties

AbstractWe construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the ‘tropical motivic nearby fiber’. This invariant specializes in the schön case to the Hodge–Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge–Deligne polynomial in the cases of schön hypersurfaces and matroidal tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration.

EXT ALGEBRA OF NICHOLS ALGEBRAS OF CARTAN TYPE A2

2013 ◽  
Vol 12 (04) ◽  
pp. 1250191
Author(s):  
XIAOLAN YU ◽  
YINHUO ZHANG
Keyword(s):  
Spectral Sequence ◽  
Hopf Algebras ◽  
Type A ◽  
Serre Spectral Sequence ◽  
Nichols Algebra ◽  
Cartan Type ◽  
Dynkin Diagrams ◽  
Full Structure ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras

We give the full structure of the Ext algebra of any Nichols algebra of Cartan type A2 by using the Hochschild–Serre spectral sequence. As an application, we show that the pointed Hopf algebras [Formula: see text] with Dynkin diagrams of type A, D, or E, except for A1 and A1 × A1 with the order NJ > 2 for at least one component J, are wild.

scholarly journals Phylogenetic relationships within the family Halobacteriaceae inferred from rpoB′ gene and protein sequences

2007 ◽  
Vol 57 (10) ◽  
pp. 2289-2295 ◽  
Author(s):  
Madalin Enache ◽  
Takashi Itoh ◽  
Tadamasa Fukushima ◽  
Ron Usami ◽  
Lucia Dumitru ◽  
...  
Keyword(s):  
16S Rrna ◽  
Molecular Marker ◽  
16S Rrna Gene ◽  
Phylogenetic Trees ◽  
Protein Sequences ◽  
Rpob Gene ◽  
Rrna Gene ◽  
Gene Sequences ◽  
16S Rrna Gene Sequences ◽  
The Family

In order to clarify the current phylogeny of the haloarchaea, particularly the closely related genera that have been difficult to sort out using 16S rRNA gene sequences, the DNA-dependent RNA polymerase subunit B′ gene (rpoB′) was used as a complementary molecular marker. Partial sequences of the gene were determined from 16 strains of the family Halobacteriaceae. Comparisons of phylogenetic trees inferred from the gene and protein sequences as well as from corresponding 16S rRNA gene sequences suggested that species of the genera Natrialba, Natronococcus, Halobiforma, Natronobacterium, Natronorubrum, Natrinema/Haloterrigena and Natronolimnobius formed a monophyletic group in all trees. In the RpoB′ protein tree, the alkaliphilic species Natrialba chahannaoensis, Natrialba hulunbeirensis and Natrialba magadii formed a tight group, while the neutrophilic species Natrialba asiatica formed a separate group with species of the genera Natronorubrum and Natronolimnobius. Species of the genus Natronorubrum were split into two groups in both the rpoB′ gene and protein trees. The most important advantage of the use of the rpoB′ gene over the 16S rRNA gene is that sequences of the former are highly conserved amongst species of the family Halobacteriaceae. All sequences determined so far can be aligned unambiguously without any gaps. On the other hand, gaps are necessary at 49 positions in the inner part of the alignment of 16S rRNA gene sequences. The rpoB′ gene and protein sequences can be used as an excellent alternative molecular marker in phylogenetic analysis of the Halobacteriaceae.

scholarly journals Multi-cluster complexes

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Cesar Ceballos ◽  
Jean-Philippe Labbé ◽  
Christian Stump
Keyword(s):  
Coxeter Groups ◽  
Type A ◽  
Simplicial Complexes ◽  
Cluster Complex ◽  
Geometric Properties ◽  
Cluster Complexes ◽  
Combinatorial Description ◽  
Compatibility Relation ◽  
Positive Roots ◽  
International Audience

International audience We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex. Nous présentons une famille de complexes simpliciaux appelés \emphcomplexes des multi-amas. Ces complexes généralisent le concept de complexes des amas et étendent la notion de multi-associaèdre de type ${A}$ et ${B}$ aux groupes de Coxeter finis. Nous étudions des propriétés combinatoires et géométriques de ces objets et, en particulier nous fournissons une description combinatoire simple de la relation de compatibilité sur l'ensemble des racines presque positives du complexe des amas.

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