Tits polygons

2022 ◽  
Vol 275 (1352) ◽  
Author(s):  
Bernhard Mühlherr ◽  
Richard Weiss ◽  
Holger Petersson
Keyword(s):  
Rational Points ◽  
Jordan Algebras ◽  
Parabolic Subgroups ◽  
Characteristic P ◽  
Arbitrary Group ◽  
Content Type ◽  
Defining Relations ◽  
Moufang Condition ◽  
Unique Vertex ◽  
Natural Way

We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank  2 2 ” presentation for the group of F F -rational points of an arbitrary exceptional simple group of F F -rank at least  4 4 and to determine defining relations for the group of F F -rational points of an an arbitrary group of F F -rank  1 1 and absolute type D 4 D_4 , E 6 E_6 , E 7 E_7 or E 8 E_8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.

Probabilistic convergence transformation groups

2018 ◽  
Vol 68 (6) ◽  
pp. 1447-1464 ◽  
Author(s):  
T. M. G. Ahsanullah ◽  
Gunther Jäger
Keyword(s):  
Transformation Group ◽  
Convergence Space ◽  
Probabilistic Metric Space ◽  
Arbitrary Group ◽  
Convergence Group ◽  
Group Of Homeomorphisms ◽  
Convergence Structure ◽  
Probabilistic Metric ◽  
Probabilistic Convergence Space ◽  
Natural Way

Abstract We introduce a notion of a probabilistic convergence transformation group, and present various natural examples including quotient probabilistic convergence transformation group. In doing so, we construct a probabilistic convergence structure on the group of homeomorphisms and look into a probabilistic convergence group that arises from probabilistic uniform convergence structure on function spaces. Given a probabilistic convergence space, and an arbitrary group, we construct a probabilistic convergence transformation group. Introducing a notion of a probabilistic metric convergence transformation group on a probabilistic metric space, we obtain in a natural way a probabilistic convergence transformation group.

scholarly journals Eisenstein Series Arising from Jordan Algebras

2019 ◽  
Vol 72 (1) ◽  
pp. 183-201 ◽  
Author(s):  
Marcela Hanzer ◽  
Gordan Savin
Keyword(s):  
Eisenstein Series ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Algebra Structure ◽  
Parabolic Subgroups ◽  
Automorphic Representations ◽  
Unipotent Radicals ◽  
Maximal Parabolic Subgroups

AbstractWe describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

Introduction to the Schwartz Space of T\G

1989 ◽  
Vol 41 (2) ◽  
pp. 285-320 ◽  
Author(s):  
W. Casselman
Keyword(s):  
Reductive Group ◽  
Rational Points ◽  
Schwartz Space ◽  
Parabolic Subgroups ◽  
Arithmetic Groups ◽  
Similar Question ◽  
Arithmetic Subgroup ◽  
Cohomology Of Arithmetic Groups ◽  
Image Position

Let G be the group of R-rational points on a reductive group defined over Q and T an arithmetic subgroup. The aim of this paper is to describe in some detail the Schwartz space (whose definition I recall in Section 1) and in particular to explain a decomposition of this space into constituents parametrized by the T-associate classes of rational parabolic subgroups of G. This is analogous to the more elementary of the two well known decompositions of L2 (T\G) in [20](or [17]), and a proof of something equivalent was first sketched by Langlands himself in correspondence with A. Borel in 1972. (Borel has given an account of this in [8].)Langlands’ letter was in response to a question posed by Borel concerning a decomposition of the cohomology of arithmetic groups, and the decomposition I obtain here was motivated by a similar question, which is dealt with at the end of the paper.

Strongly Lie nilpotent group algebras of index at most 8

2014 ◽  
Vol 13 (07) ◽  
pp. 1450044 ◽  
Author(s):  
Harish Chandra ◽  
Meena Sahai
Keyword(s):  
Nilpotent Group ◽  
Group Algebras ◽  
Characteristic P ◽  
Arbitrary Group

Let K be a field of characteristic p > 0 and let G be an arbitrary group. In this paper, we classify group algebras KG which are strongly Lie nilpotent of index at most 8. We also show that for k ≤ 6, KG is strongly Lie nilpotent of index k if and only if it is Lie nilpotent of index k.

The Number of Rational Points of a Family of Algebraic Varieties over Finite Fields

2017 ◽  
Vol 24 (04) ◽  
pp. 705-720 ◽  
Author(s):  
Shuangnian Hu ◽  
Junyong Zhao
Keyword(s):  
Number Theory ◽  
Normal Form ◽  
Finite Fields ◽  
Rational Points ◽  
System Of Equations ◽  
Algebraic Varieties ◽  
Natural Conditions ◽  
Characteristic P ◽  
Exponent Matrix ◽  
Positive Integers

Let 𝔽q stand for the finite field of odd characteristic p with q elements (q = pn, n ∈ ℕ) and [Formula: see text] denote the set of all the nonzero elements of 𝔽q. Let m and t be positive integers. By using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over [Formula: see text] where the integers t > 0, r0 = 0 < r1 < r2 < ⋯ < rt, 1 ≤ n1 < n2 <, ⋯ < nt and 0 ≤ j ≤ t − 1, bk ∊ 𝔽q, ak,i ∊ [Formula: see text] (k = 1, …, m, i = 1, …, rt), and the exponent of each variable is a positive integer. Further, under some natural conditions, we arrive at an explicit formula for the number of 𝔽q-rational points on the above variety. It extends the results obtained previously by Wolfmann, Sun, Wang, Hong et al. Our result gives a partial answer to an open problem raised in [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015) 135–153].

scholarly journals Followers’ reactions to influencers’ Instagram posts

2020 ◽  
Vol 24 (1) ◽  
pp. 37-54
Author(s):  
Daniel Belanche ◽  
Marta Flavián ◽  
Sergio Ibáñez-Sánchez
Keyword(s):  
Information Search ◽  
Product Information ◽  
Product Involvement ◽  
Content Type ◽  
Positive Behaviors ◽  
Customer Interaction ◽  
Search For Information ◽  
Marketing Actions ◽  
Practical Implications ◽  
Natural Way

Purpose The purpose of this study is to analyze how positive behaviors toward influencers (customer interaction) and promoted products (looking for product information) can be achieved, taking into account influencer–product fit, in a fashion marketing campaign. In addition, account following and product involvement are examined as possible moderators in these relationships. Design/methodology/approach The data were gathered from online participants. The participants were Instagram users who already knew a popular influencer on the platform. The experimental design manipulated the types of picture posted by the influencer to observe customers’ reactions in terms of intention to interact with the influencer’s account and to look for further information about promoted products. Findings The authors’ findings suggested that influencer–product matches in posts on Instagram encourage users to search for information about promoted products but do not affect their intention to interact with influencers’ accounts. Nevertheless, customers’ reactions toward an influencer’s posts differ based on whether they are followers of the influencer and whether they are highly or lowly involved with the promoted product. Practical implications Both brands and influencers should properly manage influencer marketing actions. Brands should control influencers’ audiences and their involvement with featured products so that they are seen to promote them in a natural way. Influencers should endorse branded products that fit their own style; this will increase the interaction on their accounts. Originality/value This research contributes to a better understanding of how users can be encouraged to undertake positive online actions as regards influencers (interaction with their accounts) and promoted products (information search) in influencer marketing campaigns.

How to be a host leader

2015 ◽  
Vol 29 (3) ◽  
pp. 15-17
Author(s):  
Mark McKergow
Keyword(s):  
Design Methodology ◽  
Content Type ◽  
Leading Position ◽  
Natural Way

Purpose – To look at what we do, as hosts. Design/methodology/approach – The paper looks at what we do as hosts, what it means to lead and leadership as engagement. Findings – Host leadership offers an alternative. Leadership seems to put the focus on the leader. The authors think this is a mistake. Leading is about a relationship – between the leader and the others. Originality/value – Host leadership is a natural way to take a leading position, in a manner that draws in others. This builds engagement, leading to performance and results.

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