scholarly journals On the ${\cal S}_{n}$-Modules Generated by Partitions of a Given Shape

10.37236/835 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Daniel Kane ◽  
Steven Sivek
Keyword(s):  
Young Diagram ◽  
Module Homomorphism

Given a Young diagram $\lambda$ and the set $H^{\lambda}$ of partitions of $\{1,2,\dots$, $|\lambda|\}$ of shape $\lambda$, we analyze a particular ${\cal S}_{|\lambda|}$-module homomorphism ${\Bbb Q}H^{\lambda}\to{\Bbb Q}H^{\lambda'}$ to show that ${\Bbb Q}H^{\lambda}$ is a submodule of ${\Bbb Q}H^{\lambda'}$ whenever $\lambda$ is a hook $(n,1,1,\dots,1)$ with $m$ rows, $n\geq m$, or any diagram with two rows.

scholarly journals NILPOTENT ORBITS AND CODIMENSION-2 DEFECTS OF 6d${\mathcal N} = (2, 0)$ THEORIES

2013 ◽  
Vol 28 (03n04) ◽  
pp. 1340006 ◽  
Author(s):  
OSCAR CHACALTANA ◽  
JACQUES DISTLER ◽  
YUJI TACHIKAWA
Keyword(s):  
Lie Algebra ◽  
Young Diagram ◽  
Central Charge ◽  
Outer Automorphism ◽  
Natural Generalization ◽  
Nilpotent Orbits ◽  
Coulomb Branch ◽  
Local Properties

We study the local properties of a class of codimension-2 defects of the 6d [Formula: see text] theories of type J = A, D, E labeled by nilpotent orbits of a Lie algebra [Formula: see text], where [Formula: see text] is determined by J and the outer-automorphism twist around the defect. This class is a natural generalization of the defects of the six-dimensional (6d) theory of type SU (N) labeled by a Young diagram with N boxes. For any of these defects, we determine its contribution to the dimension of the Higgs branch, to the Coulomb branch operators and their scaling dimensions, to the four-dimensional (4d) central charges a and c and to the flavor central charge k.

scholarly journals Search for Young diagrams with large dimensions

2019 ◽  
pp. 33-43
Author(s):  
Vasilii S. Duzhin ◽  
◽  
Anastasia A. Chudnovskaya ◽  
Keyword(s):  
Symmetric Group ◽  
Young Diagram ◽  
Numerical Experiments ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Maximum Dimension ◽  
Asymptotic Combinatorics

Search for Young diagrams with maximum dimensions or, equivalently, search for irreducible representations of the symmetric group $S(n)$ with maximum dimensions is an important problem of asymptotic combinatorics. In this paper, we propose algorithms that transform a Young diagram into another one of the same size but with a larger dimension. As a result of massive numerical experiments, the sequence of $10^6$ Young diagrams with large dimensions was constructed. Furthermore, the proposed algorithms do not change the first 1000 elements of this sequence. This may indicate that most of them have the maximum dimension. It has been found that the dimensions of all Young diagrams of the resulting sequence starting from the 75778th exceed the dimensions of corresponding diagrams of the greedy Plancherel sequence.

scholarly journals Hook lengths in a skew Young diagram

10.37236/1309 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Young Diagram ◽  
Character Degrees ◽  
Young Diagrams ◽  
Inductive Proof

Regev and Vershik (Electronic J. Combinatorics 4 (1997), #R22) have obtained some properties of the set of hook lengths for certain skew Young diagrams, using asymptotic calculations of character degrees. They also conjectured a stronger form of one of their results. We give a simple inductive proof of this conjecture. Very recently, Regev and Zeilberger (Annals of Combinatorics, to appear) have independently proved this conjecture.

On the Modular Representations of the Symmetric Group

1954 ◽  
Vol 6 ◽  
pp. 486-497 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Young Diagram ◽  
Modular Representation ◽  
Modular Representations ◽  
Modular Representation Theory

The study of the modular representation theory of the symmetric group has been greatly facilitated lately by the introduction of the graph (9, III ), the q-graph (5) and the hook-graph (4) of a Young diagram [λ]. In the present paper we seek to coordinate these ideas and relate them to the r-inducing and restricting processes (9, II ).

On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules

2020 ◽  
Vol 27 (1) ◽  
pp. 103-110
Author(s):  
Shahram Motmaen ◽  
Ahmad Yousefian Darani
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Finitely Generated ◽  
Module Homomorphism ◽  
Prime Submodule ◽  
Bezout Module

AbstractIn this paper, we introduce some classes of R-modules that are closely related to the classes of Prüfer, Dedekind and Bezout modules. Let R be a commutative ring with identity and set\mathbb{H}=\bigl{\{}M\mid M\text{ is an }R\text{-module and }\mathrm{Nil}(M)% \text{ is a divided prime submodule of }M\bigr{\}}.For an R-module {M\in\mathbb{H}}, set {T=(R\setminus Z(R))\cap(R\setminus Z(M))}, {\mathfrak{T}(M)=T^{-1}M} and {P=(\mathrm{Nil}(M):_{R}M)}. In this case, the mapping {\Phi:\mathfrak{T}(M)\to M_{P}} given by {\Phi(x/s)=x/s} is an R-module homomorphism. The restriction of Φ to M is also an R-module homomorphism from M into {M_{P}} given by {\Phi(x)=x/1} for every {x\in M}. A nonnil submodule N of M is said to be Φ-invertible if {\Phi(N)} is an invertible submodule of {\Phi(M)}. Moreover, M is called a Φ-Prüfer module if every finitely generated nonnil submodule of M is Φ-invertible. If every nonnil submodule of M is Φ-invertible, then we say that M is a Φ-Dedekind module. Furthermore, M is said to be a Φ-Bezout module if {\Phi(N)} is a principal ideal of {\Phi(M)} for every finitely generated submodule N of the R-module M. The paper is devoted to the study of the properties of Φ-Prüfer, Φ-Dedekind and Φ-Bezout R-modules.

scholarly journals On Partitions and Arf Semigroups

2019 ◽  
Vol 17 (1) ◽  
pp. 343-355
Author(s):  
Nesrin Tutaş
Keyword(s):  
Positive Integer ◽  
Young Diagram ◽  
Combinatorial Properties ◽  
Arf Semigroups

Abstract In this study we examine some combinatorial properties of the Arf semigroup. In previous work, the author and Karakaş, Gümüşbaş defined an Arf partition of a positive integer n. Here, we continue this work and give new results on Arf partitions. In particular, we analyze the relation among an Arf partition, its Young dual diagram, and the corresponding rational Young diagram. Additionally, this study contains some results that present the relations between partitions and Arf semigroup polynomials.

scholarly journals Mixed Symmetry-Tipe (k,1) Massless Tensor Fields. Consistent Interactions Of Dual Linearized Gravity

2012 ◽  
Vol 56 (1) ◽  
pp. 106-111
Author(s):  
C. Bizdadea ◽  
S. O. Saliu ◽  
M. Toma
Keyword(s):  
Master Equation ◽  
Young Diagram ◽  
Tensor Field ◽  
Local Cohomology ◽  
Tensor Fields ◽  
Linearized Gravity ◽  
Mixed Symmetry

AbstractA particular case of interactions of a single massless tensor field with the mixed symmetry corresponding to a two-column Young diagram (k,1) with k=4, dual to linearized gravity in D=7, is considered in the context of: self-couplings, cross-interactions with a Pauli-Fierz field, cross-couplings to purely matter theories, and interactions with an Abelian 1-form. The general approach relies on the deformation of the solution to the master equation from the antifield-BRST formalism by means of the local cohomology of the BRST differential.

scholarly journals An algorithm which generates linear extensions for a generalized Young diagram with uniform probability

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Kento Nakada ◽  
Shuji Okamura
Keyword(s):  
Young Diagram ◽  
Linear Extensions ◽  
Reduced Decompositions ◽  
Uniform Probability ◽  
Hook Formula ◽  
International Audience

International audience The purpose of this paper is to present an algorithm which generates linear extensions for a generalized Young diagram, in the sense of D. Peterson and R. A. Proctor, with uniform probability. This gives a proof of a D. Peterson's hook formula for the number of reduced decompositions of a given minuscule elements. \par Le but de ce papier est présenter un algorithme qui produit des extensions linéaires pour un Young diagramme généralisé dans le sens de D. Peterson et R. A. Proctor, avec probabilité constante. Cela donne une preuve de la hook formule d'un D. Peterson pour le nombre de décompositions réduites d'un éléments minuscules donné.

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