scholarly journals Application of graph combinatorics to rational identities of type $A$

10.37236/234 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Adrien Boussicault ◽  
Valentin Féray
Keyword(s):  
Rational Function ◽  
Algebraic Property ◽  
Type A ◽  
Combinatorial Property ◽  
Linear Extensions ◽  
Oriented Graphs

To a word $w$, we associate the rational function $\Psi_w = \prod (x_{w_i} - x_{w_{i+1}})^{-1}$. The main object, introduced by C. Greene to generalize identities linked to the Murnaghan-Nakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph $G$. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain).

scholarly journals Application of graph combinatorics to rational identities of type $A^\ast$

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Adrien Boussicault ◽  
Valentin Féray
Keyword(s):  
Rational Function ◽  
Algebraic Property ◽  
Type A ◽  
Combinatorial Property ◽  
Linear Extensions ◽  
Oriented Graphs ◽  
International Audience

International audience To a word $w$, we associate the rational function $\Psi_w = \prod (x_{w_i} - x_{w_{i+1}})^{-1}$. The main object, introduced by C. Greene to generalize identities linked to Murnaghan-Nakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph $G$. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain). À un mot $w$, nous associons la fonction rationnelle $\Psi_w = \prod (x_{w_i} - x_{w_{i+1}})^{-1}$. L'objet principal, introduit par C. Greene pour généraliser des identités rationnelles liées à la règle de Murnaghan-Nakayama, est une somme de ses images par certaines permutations des variables. Les ensembles de permutations considérés sont les extensions linéaires des graphes orientés. Nous expliquons comment calculer cette fonction rationnelle à partir de la combinatoire du graphe $G$. Nous établissons ensuite un lien entre une propriété algébrique de la fonction rationnelle (la factorisation du numérateur) et une propriété combinatoire du graphe (l'existence d'une chaîne le déconnectant).

scholarly journals Combinatorics of Exceptional Sequences in Type A

10.37236/6251 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Alexander Garver ◽  
Kiyoshi Igusa ◽  
Jacob P. Matherne ◽  
Jonah Ostroff
Keyword(s):  
Dynkin Diagram ◽  
Type A ◽  
Linear Extensions ◽  
Quiver Representations ◽  
Noncrossing Partitions ◽  
Underlying Graph ◽  
Exceptional Sequences

Exceptional sequences are certain sequences of quiver representations.  We introduce a class of objects called strand diagrams and use these to classify exceptional sequences of representations of a quiver whose underlying graph is a type $\mathbb{A}_n$ Dynkin diagram. We also use variations of these objects to classify $c$-matrices of such quivers, to interpret exceptional sequences as linear extensions of explicitly constructed posets, and to give a simple bijection between exceptional sequences and certain saturated chains in the lattice of noncrossing partitions. 

scholarly journals A combinatorial model for exceptional sequences in type A

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Alexander Garver ◽  
Jacob P. Matherne
Keyword(s):  
Type A ◽  
Linear Extensions ◽  
Quiver Representations ◽  
Noncrossing Partitions ◽  
Labeled Trees ◽  
Combinatorial Model ◽  
International Audience ◽  
Exceptional Sequences ◽  
Nous Utilisons

International audience Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya’s work) to classify exceptional sequences of representations of $Q$, the linearly ordered quiver with $n$ vertices. We also show how to use variations of this model to classify $c$-matrices of $Q$, to interpret exceptional sequences as linear extensions, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. In the case of $c$-matrices, we also give an interpretation of $c$-matrix mutation in terms of our noncrossing trees with directed edges. Les suites exceptionnelles sont certaines suites ordonnées de représentations de carquois. Nous utilisons des arbres aux arêtes étiquetés et aux sommets dans le bord d’un disque (expansion sur le travail de T. Araya) pour classifier les suites exceptionnelles de représentations du carquois linéairement ordonné à $n$ sommets. Nous exploitons des variations de ce modèle pour classifier les $c$-matrices dudit carquois, pour interpréter les suites exceptionnelles comme des extensions linéaires, et pour donner une bijection élémentaire entre les suites exceptionnelles et certaines chaînes dans le réseau des partitions sans croisement. Dans le cas des $c$-matrices, nous donnons également une interprétation de la mutation des $c$-matrices en termes des arbres sans croisement aux arêtes orientés.

Biological and Morphological Studies on Low-Leukemia-Strain Mice with Hodgkin's-Like Neoplasia Induced by Serial Cell-Free Transmission of Leukemic Organs of SJL/J Strain Mice

1968 ◽  
Vol 26 ◽  
pp. 210-211
Author(s):  
S. Fujinaga ◽  
K. Maruyama ◽  
C.W. Williams ◽  
K. Sekhri ◽  
L. Dmochowski
Keyword(s):  
Tissue Culture ◽  
Type A ◽  
Reticulum Cell ◽  
Type B ◽  
Viral Origin ◽  
Serial Transfer ◽  
Strain Mouse ◽  
Morphological Studies ◽  
Cell Neoplasm ◽  
Free Material

Yumoto and Dmochowski (Cancer Res.27, 2098 (1967)) reported the presence of mature and immature type C leukemia virus particles in leukemic organs and tissues such as lymph nodes, spleen, thymus, liver, and kidneys of SJL/J strain mice with Hodgki's-like disease or reticulum cell neoplasm (type B). In an attempt to ascertain the possibility that this neoplasia may be of viral origin, experiments with induction and transmission of this neoplasm were carried out using cell-free extracts of leukemic organs from an SJL/J strain mouse with spontaneous disease.It has been possible to induce the disease in low-leukemia BALB/c and C3HZB strain mice and serially transfer the neoplasia by cell-free extracts of leukemic organs of these mice. Histological examination revealed the neoplasia to be of either reticulum cell-type A or type B. Serial transfer is now in its fifth passage. In addition leukemic spleen from another SJL/J strain mouse with spontaneous reticulum cell neoplasm (type A) was set up in tissue culture and is now in its 141st serial passage in vitro. Preliminary results indicate that cell-free material of 39th tissue culture passage can reproduce neoplasia in BALB/c mice.

Hepatitis A Antigen in Liver, Bile and Stool

1975 ◽  
Vol 33 ◽  
pp. 340-341
Author(s):  
D.R. Jackson ◽  
J.H. Hoofnagle ◽  
A.N. Schulman ◽  
J.L. Dienstag ◽  
R.H. Purcell ◽  
...  
Keyword(s):  
Hepatitis A ◽  
Liver Enzymes ◽  
Hepatitis A Virus ◽  
Type A ◽  
Initial Study ◽  
Virus Like Particle ◽  
Elevated Liver Enzymes ◽  
Ag Particles ◽  
Ag Particle ◽  
Human Stool

Using immune electron microscopy Feinstone et. al. demonstrated the presence of a 27 nm virus-like particle in acute-phase stools of patients with viral hepatitis, type A, These hepatitis A antigen (HA Ag) particles were aggregated by convalescent serum from patients with type A hepatitis but not by pre-infection serum. Subsequently Dienstag et. al. and Maynard et. al. produced acute hepatitis in chimpanzees by inoculation with human stool containing HA Ag. During the early acute disease, virus like particles antigenically, morphologically and biophysically identical to the human HA Ag particle were found in chimpanzee stool. Recently Hilleman et. al. have described similar particles in liver and serum of marmosets infected with hepatitis A virus (HAV). We have investigated liver, bile and stool from chimpanzees and marmosets experimentally infected with HAV. In an initial study, a chimpanzee (no.785) inoculated with HA Ag-containing stool developed elevated liver enzymes 21 days after exposure.

Detection of Viral Antigens in Mouse and Rat Tumor Cells by Immunoelectron Microscopy Following Periodate-Lysine-Paraformaldehyde (PLP) Fixation

1976 ◽  
Vol 34 ◽  
pp. 252-253
Author(s):  
Y. Ohtsuki ◽  
G. Seman ◽  
J. M. Bowen ◽  
M. Scanlon ◽  
L. Dmochowski
Keyword(s):  
Immunoelectron Microscopy ◽  
Mammary Tumor ◽  
Type A ◽  
Rabbit Serum ◽  
Virus Particles ◽  
Viral Antigens ◽  
Culture Cells ◽  
Mouse Mammary Tumor ◽  
Tumor Virus ◽  
Immunoperoxidase Labeling

Recently, periodate-lysine-paraformaldehyde (PLP) fixation was reported for immunoelectron microscopy (1). In PLP fixation, carbohydrates are oxidized by periodate and cross-linked by lysine; paraformaldehyde stabilizes proteins and lipids. By using PLP fixation, intracytoplasmic type A viral antigens have been previously demonstrated by immunoperoxidase labeling (2). In the present study, PLP fixation has been applied for the detection of the same antigens in mouse mammary tumor culture cells by both immunoferritin and immunoperoxidase methods. Rabbit anti-intracytoplasmic type A virus serum (anti-A), kindly provided by Dr. M. Muller (3), rabbit anti-strain A mouse mammary tumor virus (anti-MMTV) and preimmune rabbit serum as control were used to detect viral antigens in cells of C3H/HeJ strain mouse mammary tumor culture. Attempts have been also made to demonstrate peroxidase labeling of type C virus particles in frozen sections of an SD-MSV-induced NZB rat bone tumor tissue by rabbit anti-MuLV serum.

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