scholarly journals Some Results on the Cohomology of Line Bundles on the Three Dimensional Flag Variety

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 295
Author(s):  
Muhammad Anwar
Keyword(s):  
Borel Subgroup ◽  
Three Dimensional ◽  
Linear Algebraic Group ◽  
Flag Variety ◽  
Line Bundles ◽  
Closed Field ◽  
Two Dimensional ◽  
Prime Characteristic ◽  
Simply Connected ◽  
The Given

Let k be an algebraically closed field of prime characteristic and let G be a semisimple, simply connected, linear algebraic group. It is an open problem to find the cohomology of line bundles on the flag variety G / B , where B is a Borel subgroup of G. In this paper we consider this problem in the case of G = S L 3 ( k ) and compute the cohomology for the case when ⟨ λ , α ∨ ⟩ = − p n a − 1 , ( 1 ≤ a ≤ p , n > 0 ) or ⟨ λ , α ∨ ⟩ = − p n − r , ( r ≥ 2 , n ≥ 0 ) . We also give the corresponding results for the two dimensional modules N α ( λ ) . These results will help us understand the representations of S L 3 ( k ) in the given cases.

scholarly journals On the Steinberg Character of a Finite Simple Group of Lie Type

1971 ◽  
Vol 12 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Bhama Srinivasan
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Reductive Algebraic Group ◽  
Simple Algebraic Group ◽  
Simply Connected ◽  
Groups Of Lie Type ◽  
Steinberg Character

Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup Gσ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of Gσ with the following properties. χ vanishes at all elements of Gσ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ±n(x) where n(x)is the order of a Sylow p-subgroup of (ZG(x))σ (ZG(x) is the centraliser of x in G). If G is simple he has, in [6], identified the possible groups Gσ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K, that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of Gσ. This expression for χ gives an explanation for the occurence of n(x) in the formula for χ (x), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group ZG(x).

Design of a Two-Piece Brassiere Cup From its Data Points Toward its Automation

2020 ◽  
Author(s):  
Kotaro Yoshida ◽  
Hidefumi Wakamatsu ◽  
Eiji Morinaga ◽  
Takahiro Kubo
Keyword(s):  
Three Dimensional ◽  
Developable Surface ◽  
Two Dimensional ◽  
Trial And Error ◽  
Design Efficiency ◽  
Developable Surfaces ◽  
Data Points ◽  
Dimensional Shape ◽  
The Given ◽  
Three Dimensional Shape

Abstract A method to design the two-dimensional shapes of patterns of two piece brassiere cup is proposed when its target three-dimensional shape is given as a cloud of its data points. A brassiere cup consists of several patterns and their shapes are designed by repeatedly making a paper cup model and checking its three-dimensional shape. For improvement of design efficiency of brassieres, such trial and error must be reduced. As a cup model for check is made of paper not cloth, it is assumed that the surface of the model is composed of several developable surfaces. When two lines that consist in the developable surface are given, the surface can be determined. Then, the two-piece brassiere cup can be designed by minimizing the error between the surface and given data points. It was mathematically verified that the developable surface calculated by our propose method can reproduce the given data points which is developable surface.

scholarly journals GENERALIZED HILBERT–KUNZ FUNCTION IN GRADED DIMENSION 2

2016 ◽  
Vol 230 ◽  
pp. 1-17 ◽  
Author(s):  
HOLGER BRENNER ◽  
ALESSIO CAMINATA
Keyword(s):  
Rational Number ◽  
Bounded Function ◽  
Closed Field ◽  
Two Dimensional ◽  
Prime Characteristic ◽  
Algebraically Closed Field ◽  
Graded Module ◽  
Dimension 2

We prove that the generalized Hilbert–Kunz function of a graded module $M$ over a two-dimensional standard graded normal $K$-domain over an algebraically closed field $K$ of prime characteristic $p$ has the form $gHK(M,q)=e_{gHK}(M)q^{2}+\unicode[STIX]{x1D6FE}(q)$, with rational generalized Hilbert–Kunz multiplicity $e_{gHK}(M)$ and a bounded function $\unicode[STIX]{x1D6FE}(q)$. Moreover, we prove that if $R$ is a $\mathbb{Z}$-algebra, the limit for $p\rightarrow +\infty$ of the generalized Hilbert–Kunz multiplicity $e_{gHK}^{R_{p}}(M_{p})$ over the fibers $R_{p}$ exists, and it is a rational number.

Solvable Subgroups and their Lie Algebras in Characteristic p

1979 ◽  
Vol 31 (2) ◽  
pp. 308-311
Author(s):  
David J. Winter
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Maximal Torus ◽  
Borel Subgroup ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Connected Component ◽  
Characteristic P ◽  
Maximal Tori ◽  
Commutative Subalgebras

1. Introduction. Throughout this paper, G is a connected linear algebraic group over an algebraically closed field whose characteristic is denoted p. For any closed subgroup H of G, denotes the Lie algebra of H and H0 denotes the connected component of the identity of H.A Borel subalgebra of is the Lie algebra of some Borel subgroup B of G. A maximal torus of is the Lie algebra of some maximal torus T of G. In [4], it is shown that the maximal tori of are the maximal commutative subalgebras of consisting of semisimple elements, and the question was raised in § 14.3 as to whether the set of Borel subalgebras of is the set of maximal triangulable subalgebras of .

On Three and Two-Dimensional Disturbances of Pipe Flow

1960 ◽  
Vol 27 (3) ◽  
pp. 381-389 ◽  
Author(s):  
Kurt Spielberg ◽  
Hans Timan
Keyword(s):  
Order Differential Equation ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Main Flow ◽  
Two Dimensional ◽  
Straight Pipe ◽  
Special Cases ◽  
Set Up ◽  
The Stability ◽  
The Given

A system of ordinary, coupled differential equations is set up for three-dimensional disturbances of Poiseuille flow in a straight pipe of circular cross section. The commonly treated equations are shown to be special cases arising from particular assumptions. It is shown that in the nonviscous, and therefore also in the general case, there exists, in contrast to the analogous problem in Cartesian co-ordinates, no transformation reducing the given problem to a two-dimensional one. A fourth-order differential equation is derived for disturbances independent of the direction of the main flow. The solutions, which are obtained, show that those two-dimensional disturbances, termed cross disturbances, decay with time and do therefore not disturb the stability of the main flow. Explicit expressions for the cross disturbances are obtained and a discussion of their nature is given.

scholarly journals Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block

2018 ◽  
Vol 21 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Donna M. Testerman ◽  
Alexandre E. Zalesski
Keyword(s):  
Algebraic Groups ◽  
Jordan Block ◽  
Linear Algebraic Group ◽  
Irreducible Representations ◽  
Closed Field ◽  
Jordan Normal Form ◽  
Unipotent Element ◽  
Content Type ◽  
Simply Connected ◽  
Graphic Content

AbstractLetGbe a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed fieldFof characteristic{p\geq 0}, and let{u\in G}be a nonidentity unipotent element. Let ϕ be a non-trivial irreducible representation ofG. Then the Jordan normal form of{\phi(u)}contains at most one non-trivial block if and only ifGis of type{G_{2}},uis a regular unipotent element and{\dim\phi\leq 7}. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I. D. Suprunenko [21].

scholarly journals On the Standard Poisson Structure and a Frobenius Splitting of the Basic Affine Space

2019 ◽  
Author(s):  
Jun Peng ◽  
Shizhuo Yu
Keyword(s):  
General Theory ◽  
Poisson Structure ◽  
Borel Subgroup ◽  
Closed Field ◽  
Poisson Geometry ◽  
Characteristic P ◽  
Frobenius Splitting ◽  
Algebraic Torus ◽  
Poisson Varieties ◽  
Simply Connected

Abstract The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$, where $G$ is a simply connected semi-simple algebraic group defined over an algebraically closed field of characteristic $p> 3$, $U$ is the uniradical of a Borel subgroup of $G$, and $\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$. Then we develop a general theory for Frobenius splittings on $\mathbb{T}$-Poisson varieties, where $\mathbb{T}$ is an algebraic torus. In particular, we prove that compatibly split subvarieties of Frobenius splittings constructed in this way must be $\mathbb{T}$-Poisson subvarieties. Lastly, we apply our general theory to construct a Frobenius splitting on $G/U$.

An Efficient Technique for Three-Dimensional Image Visualization Through Two-Dimensional Images for Medical Data

2017 ◽  
Vol 29 (1) ◽  
pp. 100-109
Author(s):  
Ganesan Gunasekaran ◽  
Meenakshisundaram Venkatesan
Keyword(s):  
Three Dimensional ◽  
Main Idea ◽  
Two Dimensional ◽  
3D Image ◽  
Image Visualization ◽  
Good Determination ◽  
The Given ◽  
Geometric Primitives ◽  
2D Images

Abstract The main idea behind this work is to present three-dimensional (3D) image visualization through two-dimensional (2D) images that comprise various images. 3D image visualization is one of the essential methods for excerpting data from given pieces. The main goal of this work is to figure out the outlines of the given 3D geometric primitives in each part, and then integrate these outlines or frames to reconstruct 3D geometric primitives. The proposed technique is very useful and can be applied to many kinds of images. The experimental results showed a very good determination of the reconstructing process of 2D images.

scholarly journals Infinitesimal Invariants in a Function Algebra

2009 ◽  
Vol 61 (4) ◽  
pp. 950-960 ◽  
Author(s):  
Rudolf Tange
Keyword(s):  
Simple Group ◽  
Mild Condition ◽  
Positive Characteristic ◽  
Function Algebra ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Reductive Groups ◽  
Regular Functions ◽  
Simply Connected ◽  
Field Of Positive Characteristic

Abstract.Let G be a reductive connected linear algebraic group over an algebraically closed field of positive characteristic and let 𝔤 be its Lie algebra. First we extend a well-known result about the Picard group of a semi-simple group to reductive groups. Then we prove that if the derived group is simply connected and 𝔤 satisfies a mild condition, the algebra K[G]𝔤 of regular functions on G that are invariant under the action of 𝔤 derived from the conjugation action is a unique factorisation domain.

scholarly journals The parabolic exotic t-structure

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Pramod Achar ◽  
Nicholas Cooney ◽  
Simon Riche
Keyword(s):  
Cotangent Bundle ◽  
Flag Variety ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Geometric Representation Theory ◽  
Coxeter Number ◽  
Derived Subgroup ◽  
International Audience ◽  
Simply Connected ◽  
Comme Application

International audience Let G be a connected reductive algebraic group over an algebraically closed field k, with simply connected derived subgroup. The exotic t-structure on the cotangent bundle of its flag variety T^*(G/B), originally introduced by Bezrukavnikov, has been a key tool for a number of major results in geometric representation theory, including the proof of the graded Finkelberg-Mirkovic conjecture. In this paper, we study (under mild technical assumptions) an analogous t-structure on the cotangent bundle of a partial flag variety T^*(G/P). As an application, we prove a parabolic analogue of the Arkhipov-Bezrukavnikov-Ginzburg equivalence. When the characteristic of k is larger than the Coxeter number, we deduce an analogue of the graded Finkelberg-Mirkovic conjecture for some singular blocks. Soit G un groupe algébrique réductif connexe sur un corps k algébriquement clos. La t-structure exotique sur le fibré cotangent de sa variété de drapeaux T^*(G/B), introduite à l'origine par Bezrukavnikov, a été un outil clé pour de nombreux résultats majeurs en théorie géométrique des représentations, en particulier la démonstration de la conjecture de Finkelberg-Mirkovic graduée. Dans cet article, nous étudions (sous de légères hypothèses techniques) une t-structure analogue sur le fibré cotangent de la variété de drapeaux partiels T^*(G/P). Comme application, nous prouvons un analogue parabolique de l'équivalence de Arkhipov-Bezrukavnikov-Ginzburg. Lorsque la caractéristique de k est supérieure au nombre de Coxeter, nous déduisons un analogue de la conjecture de Finkelberg-Mirkovic graduée pour certains blocs singuliers.

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