On the relationship between combinatorial functions and representation theory

2017 ◽  
Vol 51 (1) ◽  
pp. 22-31 ◽  
Author(s):  
A. M. Vershik ◽  
N. V. Tsilevich
Keyword(s):  
Representation Theory ◽  
The Relationship

scholarly journals Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Thomas Bliem ◽  
Dido Salazar
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Ehrhart Polynomial ◽  
International Audience ◽  
Order Polytope ◽  
Finite Partially Ordered Set ◽  
The Relationship

International audience Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras. Stanley (1986) a montré que chaque ensemble fini partiellement ordonné permet de définir deux polyèdres, le polyèdre de l'ordre et le polyèdre des cha\^ınes. Ces polyèdres ont le même polynôme de Ehrhart, bien qu'ils soient tout à fait distincts du point de vue combinatoire. On généralise ce résultat à une famille plus générale de polyèdres, construits à partir d'un ensemble partiellement ordonné ayant des entiers attachés à certains de ses éléments. Par cette construction, on explique en termes combinatoires la relation entre les polyèdres de Gelfand-Tsetlin (1950) et ceux de Feigin-Fourier-Littelmann-Vinberg (2010, 2005), qui apparaissent dans la théorie des représentations des algèbres de Lie linéaires spéciales. On utilise les polyèdres de Gelfand-Tsetlin généralisés par Berenstein et Zelevinsky (1989) afin d'obtenir des analogues (conjecturés) des polytopes de Feigin-Fourier-Littelmann-Vinberg pour les algèbres de Lie symplectiques et orthogonales impaires.

A Remark by Philip Hall

1958 ◽  
Vol 1 (1) ◽  
pp. 21-23 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Linear Group ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Linear Transformations ◽  
Modern Physics ◽  
The Right ◽  
The Relationship

The relationship between the representation theory of the full linear group GL(d) of all non-singular linear transformations of degree d over a field of characteristic zero and that of the symmetric group Sn goes back to Schur and has been expounded by Weyl in his classical groups, [4; cf also 2 and 3]. More and more, the significance of continuous groups for modern physics is being pressed on the attention of mathematicians, and it seems worth recording a remark made to the author by Philip Hall in Edmonton.As is well known, the irreducible representations of Sn are obtainable from the Young diagrams [λ]=[λ1, λ2 ,..., λr] consisting of λ1 nodes in the first row, λ2 in the second row, etc., where λ1≥λ2≥ ... ≥λr and Σ λi = n. If we denote the jth node in the ith row of [λ] by (i,j) then those nodes to the right of and below (i,j), constitute, along with the (i,j) node itself, the (i,j)-hook of length hij.

scholarly journals The role of Heisenberg group in Harmonic analysis: دور زمرة هايزنبرغ في التحليل التوافقي

2019 ◽  
Vol 3 (3) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
Mathematical Concepts ◽  
Group Fourier Transform ◽  
The Relationship

In this paper we talk about Heisenberg group, the most know example from the lie groups. After that we discuss the representation theory of this group, and the relationship between the representation theory of the Heisenberg group and the position and momentum operatorsو and momentum operators.ors. ielationship between the representation theory of the Heisenberg group and the position and momen, that shows how we will make the connection between the Heisenberg group and physics. we have considered only the Schr dinger picture. That is, all the representations we considered are realized on the Hilbert space . we define the group Fourier transform on the Heisenberg group as an operator valued function, and other facts and properties. The main aim of our research is having the formula of Schr dinger Representation that connect physics with the Heisenberg group. Depending on this Representation we will study new formulas for some mathematical concepts such us Fourier Transform and  .

Narrative Representation Theory: Identifying the human language with superstructure

2017 ◽  
Vol 19 (6) ◽  
pp. 648-672
Author(s):  
Hirokuni Masuda
Keyword(s):  
Biological Sciences ◽  
Mental Disorders ◽  
Representation Theory ◽  
Human Language ◽  
Computational System ◽  
Language Faculty ◽  
The Right ◽  
First Time ◽  
The Relationship ◽  
The Brain

Narrative Representation Theory (NRT), an evolved framework of Verse Analysis, has come into existence with the mission of explaining the operation of macro-systemic structure that could be hardwired in the brain. Based on the analyses of creoles or archetypal human languages, the theory puts forward the premise stating that the fundamental design of the human language faculty possesses the computational system for internalized discourse. The theory preserves the principles of Quint-patterning, Idea-formatting, N-ary-branching and X-numbering, complying respectively with the hierarchical orderings of constituency, the atomic elements of componentiality, the linear sequences of precedence and the specific measurement of terminal nodes. NRT tells that the macro-system of narrative superstructure must have emerged autonomously, yet links closely with the micro-system of phonology, morphology and syntax. This article explores for the first time scientific insights into the nature of human language, referring to recent research on the right cerebrum as well as on the prefrontal lobes of the brain, the relationship between mental disorders and their genetic deficiencies, and the investigations of human evolution during the period 200,000–40,000 years BP. All the converging evidence in biological sciences reinforces the hypothesis that the narrative superstructure of language faculty manifests as an inherent linguistic capacity in our mind.

scholarly journals Character theory of infinite wreath products

2005 ◽  
Vol 2005 (9) ◽  
pp. 1365-1379 ◽  
Author(s):  
Robert Boyer
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Finite Groups ◽  
Wreath Products ◽  
Character Theory ◽  
Group Algebras ◽  
Inductive Limits ◽  
Infinite Symmetric Group ◽  
The Relationship ◽  
Asymptotic Character

The representation theory of infinite wreath product groups is developed by means of the relationship between their group algebras and conjugacy classes with those of the infinite symmetric group. Further, since these groups are inductive limits of finite groups, their finite characters can be classified as limits of normalized irreducible characters of prelimit finite groups. This identification is called the “asymptotic character formula.” TheK0-invariant of the groupC∗-algebra is also determined.

scholarly journals A Riesz representation theory for completely regular Hausdorff spaces and its applications

2016 ◽  
Vol 14 (1) ◽  
pp. 474-496 ◽  
Author(s):  
Marian Nowak
Keyword(s):  
Representation Theory ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Weakly Compact ◽  
Completely Regular ◽  
Riesz Representation ◽  
Continuous Operators ◽  
Bounded Continuous Functions ◽  
The Relationship ◽  
Stieltjes Type

AbstractLet X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we study (β, || · ||F)-continuous weakly compact and unconditionally converging operators T : Cb(X, E) → F. In particular, we establish the relationship between these operators and the corresponding Borel operator measures given by the Riesz representation theorem. We obtain that if X is a k-spaceand E is reflexive, then (Cb(X, E), β) has the V property of Pełczynski.

scholarly journals Characters of infinite-dimensional quantum classical groups: BCD cases

2021 ◽  
Author(s):  
Ryosuke Sato
Keyword(s):  
Representation Theory ◽  
Character Theory ◽  
Compact Groups ◽  
Markov Semigroups ◽  
Inductive Limits ◽  
Universal Enveloping Algebras ◽  
Infinite Dimensional ◽  
Enveloping Algebras ◽  
Quantized Universal Enveloping Algebras ◽  
The Relationship

We study the character theory of inductive limits of [Formula: see text]-deformed classical compact groups. In particular, we clarify the relationship between the representation theory of Drinfeld–Jimbo quantized universal enveloping algebras and our previous work on the quantized characters. We also apply the character theory to construct Markov semigroups on unitary duals of [Formula: see text], [Formula: see text], and their inductive limits.

scholarly journals Hermite functions and special Hermite functions  depending on Heisenberg group (ℍⁿ ): دوال هرميت ودوال هرميت الخاصة اعتماداً على زمرة هايزنبرغ (ℍⁿ)

2020 ◽  
Vol 4 (2) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Representation Theory ◽  
Harmonic Oscillator ◽  
Heisenberg Group ◽  
Hermite Functions ◽  
Hermite Operator ◽  
Twisted Laplacian ◽  
Special Hermite Functions ◽  
Hermite Operators ◽  
The Relationship ◽  
Hermite Expansions

In this paper, we talk about Heisenberg group, the most known example from the lie groups. After that, we talk about the representation theory of this group, and the relationship between the representation theory of the Heisenberg group and the position and momentum operator and momentum operators (ors). relationship between the representation theory of the Heisenberg group and the position and momentum, that shows how we will make the connection between the Heisenberg group and physics. Then we introduce and study some properties of the Hermite and special Hermite functions. These functions are eigenfunctions of the Hermite and special Hermite operators, respectively. The Hermite operator is often called the harmonic oscillator and the special Hermite operator is sometimes called the twisted Laplacian. As we will later see, the two operators are directly related to the sub-laplacian on the Heisenberg group. The theory of Hermite and special Hermite expansions is intimately connected to the harmonic analysis on the Heisenberg group. They play an important role in our understanding of several problems on ℍⁿ .

Semantics

2016 ◽  
Author(s):  
David Beaver ◽  
Joey Frazee
Keyword(s):  
Natural Language Processing ◽  
Representation Theory ◽  
Language Processing ◽  
Semantic Analysis ◽  
Formal Semantics ◽  
Linguistic Meaning ◽  
Discourse Representation ◽  
Discourse Representation Theory ◽  
Hybrid Approaches ◽  
The Relationship

Formal semantics is the study of linguistic meaning using precise mathematical characterizations; this chapter introduces formal semantics to scholars and students of natural-language processing. We give simple logical representations of English sentences, and show how meanings are composed in a grammar. We then consider two more advanced issues that arise in processing texts, anaphora and temporality, using Discourse Representation Theory (DRT). Finally we discuss the relationship between deep logic-based methods for semantic analysis and shallower distributional methods that have been used in much recent NLP work, introducing some limitations of distributional methods, and hence motivating deeper or hybrid approaches.

scholarly journals Rational Dyck Paths in the Non Relatively Prime Case

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Eugene Gorsky ◽  
Mikhail Mazin ◽  
Monica Vazirani
Keyword(s):  
Representation Theory ◽  
Dyck Paths ◽  
International Audience ◽  
Springer Fibers ◽  
The Relationship ◽  
Prime Case

International audience We study the relationship between rational slope Dyck paths and invariant subsets in Z, extending the work of the first two authors in the relatively prime case. We also find a bijection between (dn, dm)–Dyck paths and d-tuples of (n, m)-Dyck paths endowed with certain gluing data. These are first steps towards understanding the relationship between the rational slope Catalan combinatorics in non relatively prime case and the geometry of affine Springer fibers and representation theory.

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