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 UN SCINDAGE DU MORPHISME DE FROBENIUS SUR L’ALGÈBRE DES DISTRIBUTIONS D’UN GROUPE RÉDUCTIF

2019 ◽  
Vol 71 (1) ◽  
pp. 197-206
Author(s):  
Michel Gros ◽  
Kaneda Masaharu
Keyword(s):  
Nous Avons ◽  
Algebraic Group ◽  
Positive Characteristic ◽  
Closed Field ◽  
Reductive Groups ◽  
Algebraically Closed Field ◽  
Frobenius Endomorphism ◽  
Simply Connected ◽  
Simplement Connexe ◽  
Field Of Positive Characteristic

Abstract Pour un groupe algébrique semi-simple simplement connexe sur un corps algébriquement clos de caractéristique positive, nous avons précédemment construit un scindage de l’endomorphisme de Frobenius sur son algèbre des distributions. Nous généralisons la construction au cas de des groupes réductifs connexes et en dégageons les corollaires correspondants. For a simply connected semisimple algebraic group over an algebraically closed field of positive characteristic we have already constructed a splitting of the Frobenius endomorphism on its algebra of distributions. We generalize the construction to the case of general connected reductive groups and derive the corresponding corollaries.

The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n − 1

2021 ◽  
pp. 2250191
Author(s):  
Merrick Cai ◽  
Daniil Kalinov
Keyword(s):  
Hilbert Series ◽  
Positive Characteristic ◽  
Closed Field ◽  
Simple Criterion ◽  
Characteristic P ◽  
Polynomial Representation ◽  
Cherednik Algebra ◽  
Rational Cherednik Algebra ◽  
Polynomial Formula ◽  
Field Of Positive Characteristic

In this paper, we study the irreducible quotient [Formula: see text] of the polynomial representation of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] over an algebraically closed field of positive characteristic [Formula: see text] where [Formula: see text]. In the [Formula: see text] case, for all [Formula: see text] we give a complete description of the polynomials in the maximal proper graded submodule [Formula: see text], the kernel of the contravariant form [Formula: see text], and subsequently find the Hilbert series of the irreducible quotient [Formula: see text]. In the [Formula: see text] case, we give a complete description of the polynomials in [Formula: see text] when the characteristic [Formula: see text] and [Formula: see text] is transcendental over [Formula: see text], and compute the Hilbert series of the irreducible quotient [Formula: see text]. In doing so, we prove a conjecture due to Etingof and Rains completely for [Formula: see text], and also for any [Formula: see text] and [Formula: see text]. Furthermore, for [Formula: see text], we prove a simple criterion to determine whether a given polynomial [Formula: see text] lies in [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text] fixed.

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).

scholarly journals The Square-Zero Basis of Matrix Lie Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1032
Author(s):  
Raúl Durán Díaz ◽  
Víctor Gayoso Martínez ◽  
Luis Hernández Encinas ◽  
Jaime Muñoz Masqué
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Characteristic ◽  
Linear Algebraic Group ◽  
Invariant Functions ◽  
Arbitrary Characteristic ◽  
Independent Invariant ◽  
Field Of Positive Characteristic

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. The class of such Lie algebras is studied in the framework of the classical Lie algebras of arbitrary characteristic. Some examples and applications are also given.

An Approach to Higher Teichmüller Spaces for General Groups

2017 ◽  
Vol 2019 (16) ◽  
pp. 4899-4949 ◽  
Author(s):  
Ian Le
Keyword(s):  
Simple Group ◽  
Moduli Spaces ◽  
Cluster Structure ◽  
Type A ◽  
Classical Groups ◽  
Reductive Groups ◽  
Local Systems ◽  
Teichmuller Spaces ◽  
Cluster Varieties ◽  
Simply Connected

Abstract Let $S$ be a surface, $G$ a simply-connected semi-simple group, and $G'$ the associated adjoint form of the group. In Fock and Goncharov [4], the authors show that the moduli spaces of framed local systems $\mathcal{X}_{G',S}$ and $\mathcal{A}_{G,S}$ have the structure of cluster varieties when $G$ had type $A$. This was extended to classical groups in Le [12]. In this article, we give a method for constructing the cluster structure for general reductive groups $G$. The method depends on being able to carry out some explicit computations, and depends on some mild hypotheses, which we state, and which we believe hold in general. These hypotheses hold when $G$ has type $G_2,$ and therefore we are able to construct the cluster structure in this case. We also illustrate our approach by rederiving the cluster structure for $G$ of type $A$. Our goals are to give some heuristics for the approach taken in Le [12], point out the difficulties that arise for more general groups, and to record some useful calculations. Forthcoming work by Goncharov and Shen gives a different approach to constructing the cluster structure on $\mathcal{X}_{G',S}$ and $\mathcal{A}_{G,S}$. We hope that some of the ideas here complement their more comprehensive work.

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 Cocharacters Associated to Nilpotent Elements of Reductive Groups

2008 ◽  
Vol 190 ◽  
pp. 105-128 ◽  
Author(s):  
Russell Fowler ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Nilpotent Element ◽  
Maximal Rank ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Reductive Groups ◽  
Algebraically Closed Field ◽  
Reductive Subgroup ◽  
Good For

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we consider particular classes of connected reductive subgroups H of G and show that the cocharacters of H that are associated to a given nilpotent element e in the Lie algebra of H are precisely the cocharacters of G associated to e that take values in H. In particular, we show that this is the case provided H is a connected reductive subgroup of G of maximal rank; this answers a question posed by J. C. Jantzen.

THE LOEWY STRUCTURE OF -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

2015 ◽  
Vol 16 (4) ◽  
pp. 887-898
Author(s):  
Noriyuki Abe ◽  
Masaharu Kaneda
Keyword(s):  
Algebraic Group ◽  
Maximal Torus ◽  
Positive Characteristic ◽  
Closed Field ◽  
Verma Modules ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Loewy Length ◽  
Frobenius Kernel ◽  
Field Of Positive Characteristic

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

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 The construction problem for Hodge numbers modulo an integer in positive characteristic

2020 ◽  
Vol 8 ◽  
Author(s):  
Remy van Dobben de Bruyn ◽  
Matthias Paulsen
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Construction Problem ◽  
Algebraically Closed Field ◽  
Serre Duality ◽  
Hodge Numbers ◽  
Field Of Positive Characteristic

Abstract Let k be an algebraically closed field of positive characteristic. For any integer $m\ge 2$ , we show that the Hodge numbers of a smooth projective k-variety can take on any combination of values modulo m, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.

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