19 results on '"Algebraic geometry"'
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2. Zero-cycles with coefficients for the second generalized symplectic involution variety of an algebra of degree 4.
- Author
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McFaddin, Patrick K.
- Subjects
- *
ALGEBRA , *MATHEMATICAL equivalence , *ALGEBRAIC geometry , *GROUP theory , *MATHEMATICS - Abstract
Abstract We compute the group of K 1 -zero-cycles on the second generalized involution variety for an algebra of degree 4 with symplectic involution. This description is given in terms of the group of multipliers of similitudes associated to the algebra with involution. Our method utilizes the framework of Chernousov and Merkurjev for computing K 1 -zero-cycles in terms of R -equivalence classes of prescribed algebraic groups. This gives a computation of K 1 -zero-cycles for some homogeneous varieties of type C 2. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
3. ZERO-ONE GENERATION LAWS FOR FINITE SIMPLE GROUPS.
- Author
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GURALNICK, ROBERT M., LIEBECK, MARTIN W., LÜBECK, FRANK, and SHALEV, ANER
- Subjects
- *
FINITE simple groups , *REPRESENTATIONS of groups (Algebra) , *ALGEBRAIC geometry , *FINITE groups , *CONJUGACY classes , *LINEAR algebraic groups - Abstract
Let G be a simple algebraic group over the algebraic closure of Fp (p prime), and let G(q) denote a corresponding finite group of Lie-type over Fq, where q is a power of p. Let X be an irreducible subvariety of Gr for some r ≥ 2. We prove a zero-one law for the probability that G(q) is generated by a random r-tuple in X(q) = X ∩ G(q)r: the limit of this probability as q increases (through values of q for which X is stable under the Frobenius morphism defining G(q)) is either 1 or 0. Indeed, to ensure that this limit is 1, one only needs G(q) to be generated by an r-tuple in X(q) for two sufficiently large values of q. We also prove a version of this result where the underlying characteristic is allowed to vary. In our main application, we apply these results to the case where r = 2 and the irreducible subvariety X = C ×D, a product of two conjugacy classes of elements of finite order in G. This leads to new results on random (2, 3)- generation of finite simple groups G(q) of exceptional Lie-type: provided G(q) is not a Suzuki group, we show that the probability that a random involution and a random element of order 3 generate G(q) tends to 1 as q→∞. Combining this with previous results for classical groups, this shows that finite simple groups (apart from Suzuki groups and PSp4(q)) are randomly (2, 3)-generated. Our tools include algebraic geometry, representation theory of algebraic groups, and character theory of finite groups of Lie-type. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
4. On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices
- Author
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Nosan, K, Pouly, A, Schmitz, S, Shirmohammadi, M, Worrell, J, Institut de Recherche en Informatique Fondamentale (IRIF (UMR_8243)), Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Department of Computer Science [Oxford], University of Oxford, Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and European Project: ERC-2019-STG 852769,ARiAT
- Subjects
FOS: Computer and information sciences ,[INFO.INFO-CC]Computer Science [cs]/Computational Complexity [cs.CC] ,Group Theory (math.GR) ,algebraic groups ,Computational Complexity (cs.CC) ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,Computer Science - Computational Complexity ,Mathematics - Algebraic Geometry ,FOS: Mathematics ,Zariski topology ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Mathematics - Group Theory ,Algebraic Geometry (math.AG) ,algebraic geometry - Abstract
International audience; We investigate the complexity of computing the Zariski closure of a finitely generated group of matrices. The Zariski closure was previously shown to be computable by Derksen, Jeandel and Koiran, but the termination argument for their algorithm appears not to yield any complexity bound. In this paper we follow a different approach and obtain a bound on the degree of the polynomials that define the closure. Our bound shows that the closure can be computed in elementary time. We describe several applications of this result, e.g., concerning quantum automata and quantum universal gates. We also obtain an upper bound on the length of a strictly increasing chain of linear algebraic groups, all of which are generated over a fixed number field.
- Published
- 2022
5. On support varieties and the Humphreys conjecture in type A.
- Author
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Hardesty, William D.
- Subjects
- *
LOGICAL prediction , *INDECOMPOSABLE modules , *MODULES (Algebra) , *ALGEBRAIC geometry , *GROUP theory - Abstract
Let G be a reductive algebraic group scheme defined over F p and let G 1 denote the Frobenius kernel of G . To each finite-dimensional G -module M , one can define the support variety V G 1 ( M ) , which can be regarded as a G -stable closed subvariety of the nilpotent cone. A G -module is called a tilting module if it has both good and Weyl filtrations. In 1997, it was conjectured by J.E. Humphreys that when p ≥ h , the support varieties of the indecomposable tilting modules align with the nilpotent orbits given by the Lusztig bijection. In this paper, we shall verify this conjecture when G = S L n + 1 and p > n + 1 . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
6. Bruhat cells which contain only regular elements.
- Author
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Rainbolt, Julianne
- Subjects
GROUP theory ,ALGEBRAIC geometry ,LINEAR algebraic groups ,SUBGROUP growth ,MATHEMATICAL mappings - Abstract
Letbe a connected reductive algebraic group defined over an algebraically closed field. Letbe a maximal torus ofand letbe a Borel subgroup ofcontaining. LetWdenote the Weyl group ofand letẇdenote an element in the preimage of the natural map fromtoW. We investigate what conditions are necessary and sufficient for the double cosets(called Bruhat cells) to contain only regular elements. An application is provided on how this information about Bruhat cells can be used to determine character values for certain Hecke algebras. [ABSTRACT FROM PUBLISHER]
- Published
- 2017
- Full Text
- View/download PDF
7. Oriented Cohomology Rings of the Semisimple Linear Algebraic Groups of Ranks 1 and 2
- Author
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Gandhi, Raj
- Subjects
Algebraic geometry ,Demazure operator ,Algebraic oriented cohomology theories ,Mathematics::Representation Theory ,Algebraic groups ,Formal group law - Abstract
In this thesis, we compute minimal presentations in terms of generators and relations for the oriented cohomology rings of several semisimple linear algebraic groups of ranks 1 and 2 over algebraically closed fields of characteristic 0. The main tools we use in this thesis are the combinatorics of Coxeter groups and formal group laws, and recent results of Calm\`es, Gille, Petrov, Zainoulline, and Zhong, which relate the oriented cohomology rings of flag varieties and semisimple linear algebraic groups to the dual of the formal affine Demazure algebra.
- Published
- 2021
- Full Text
- View/download PDF
8. GROWTH IN FINITE SIMPLE GROUPS OF LIE TYPE.
- Author
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PYBER, LÁSZLÓ and SZABÓ, ENDRE
- Subjects
- *
MATHEMATICS theorems , *LIE algebras , *ALGEBRAIC geometry , *MATHEMATICS , *ALGEBRAIC surfaces - Abstract
The article discusses the need for E(r) = O(1/r) in Theorem 2 to show how a finite simple group of Lie type can obtain a very large subgroup of G. Topics discussed include basic definitions and useful facts from algebraic geometry, the concentration of a finite set α in a closed set X as the relative size of α X in X and the establishment of the technical tool, the Spreading Theorem 40.
- Published
- 2016
- Full Text
- View/download PDF
9. A user friendly proof of Nagata's characterization of linearly reductive groups in positive characteristics.
- Author
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Kohls, Martin
- Subjects
- *
GROUP theory , *LINEAR algebraic groups , *ALGEBRAIC varieties , *ALGEBRAIC geometry , *TORUS , *MATHEMATICAL analysis - Abstract
Let K be an algebraically closed field of positive characteristic p, and G be a linear algebraic group over K. We give a user friendly proof of Nagata's theorem that every finite-dimensional rational representation of G is completely reducible if and only if the connected component G0 is a torus and p does not divide the index (G : G0). [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
10. Zero-cycles on projective varieties and the norm principle.
- Author
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Gille, Philippe and Semenov, Nikita
- Subjects
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ALGEBRAIC geometry , *ALGEBRAIC varieties , *QUADRICS , *GRASSMANN manifolds , *DIFFERENTIAL algebraic groups , *MATHEMATICAL physics - Abstract
Using the Gille--Merkurjev norm principle we compute in a uniform way the image of the degree map for quadrics (Springer's theorem), for twisted forms of maximal orthogonal Grassmannians (a theorem of Bayer-Fluckiger and Lenstra), and for E6- (a theorem of Rost) and E7-varieties. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
11. RESTRICTIONS OF SOME REPRESENTATIONS OF THE SYMPLECTIC GROUP TO SUBGROUPS OF TYPE A1.
- Author
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OSINOVSKAYA, ANNA A.
- Subjects
- *
SYMPLECTIC groups , *LINEAR algebraic groups , *GROUP theory , *ALGEBRAIC varieties , *ALGEBRAIC geometry - Abstract
Restrictions of modular irreducible representations of the symplectic algebraic group to naturally embedded long subgroups of type A1 are studied. Let ω = m1ω1 + ⋯ + mnωn be the highest weight of such representation. The composition factors of such restrictions are determined in the case of m1 + ⋯ + mn + 3 ≤ p < mn-1 + 2mn + 3 that completes the description of restrictions of classical algebraic groups to naturally embedded A1-subgroups and gives an example of a new inductive system of representations of symplectic groups that has no analogues in characteristic 0. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
12. RESTRICTIONS OF SOME REPRESENTATIONS OF THE SYMPLECTIC GROUP TO SUBGROUPS OF TYPE A1.
- Author
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OSINOVSKAYA, ANNA A.
- Subjects
SYMPLECTIC groups ,LINEAR algebraic groups ,GROUP theory ,ALGEBRAIC varieties ,ALGEBRAIC geometry - Abstract
Restrictions of modular irreducible representations of the symplectic algebraic group to naturally embedded long subgroups of type A
1 are studied. Let ω = m1 ω1 + ⋯ + mn ωn be the highest weight of such representation. The composition factors of such restrictions are determined in the case of m1 + ⋯ + mn + 3 ≤ p < mn-1 + 2mn + 3 that completes the description of restrictions of classical algebraic groups to naturally embedded A1 -subgroups and gives an example of a new inductive system of representations of symplectic groups that has no analogues in characteristic 0. [ABSTRACT FROM AUTHOR]- Published
- 2007
- Full Text
- View/download PDF
13. Canonical -dimension of algebraic groups
- Author
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Karpenko, Nikita A. and Merkurjev, Alexander S.
- Subjects
- *
ALGEBRAIC geometry , *MATHEMATICS , *NUMERICAL analysis , *ALGEBRA - Abstract
Abstract: We describe a way to compute the -relative version of the Berhuy–Reichstein canonical dimension for an arbitrary split semisimple algebraic group over an arbitrary field of an arbitrary characteristic ( is any prime integer). The canonical -dimension is computed for all split simple groups of classical types. [Copyright &y& Elsevier]
- Published
- 2006
- Full Text
- View/download PDF
14. Quantum automata and algebraic groups
- Author
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Derksen, Harm, Jeandel, Emmanuel, and Koiran, Pascal
- Subjects
- *
MACHINE theory , *SEQUENTIAL machine theory , *ALGORITHMS , *MATRICES (Mathematics) - Abstract
Abstract: We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices. [Copyright &y& Elsevier]
- Published
- 2005
- Full Text
- View/download PDF
15. Quantum automata and algebraic groups
- Author
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Pascal Koiran, Emmanuel Jeandel, and Harm Derksen
- Subjects
Quantum automata ,TheoryofComputation_COMPUTATIONBYABSTRACTDEVICES ,Dimension of an algebraic variety ,0102 computer and information sciences ,ω-automaton ,01 natural sciences ,Algebraic closure ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Real algebraic geometry ,Quantum finite automata ,Probabilistic automata ,0101 mathematics ,Mathematics ,Discrete mathematics ,Algebra and Number Theory ,010102 general mathematics ,Undecidability ,Algebraic groups ,Algebraic geometry ,Computational Mathematics ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,010201 computation theory & mathematics ,Automata theory ,Algebraic function ,Irreducible component ,Computer Science::Formal Languages and Automata Theory - Abstract
We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices.
- Published
- 2005
- Full Text
- View/download PDF
16. The Grothendieck Gamma Filtration, the Tits Algebras, and the J-invariant of a Linear Algebraic Group
- Author
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Junkins, Caroline
- Subjects
Mathematics::K-Theory and Homology ,projective homogeneous varieties ,cohomological invariants ,algebraic groups ,algebraic geometry - Abstract
Consider a semisimple linear algebraic group G over an arbitrary field F, and a projective homogeneous G-variety X. The geometry of such varieties has been a consistently active subject of research in algebraic geometry for decades, with significant contributions made by Grothendieck, Demazure, Tits, Panin, and Merkurjev, among others. An effective tool for the classification of these varieties is the notion of a cohomological (or alternatively, a motivic) invariant. Two such invariants are the set of Tits algebras of G defined by J. Tits, and the J-invariant of G defined by Petrov, Semenov, and Zainoulline. Quéguiner-Mathieu, Semenov and Zainoulline discovered a connection between these invariants, which they developed through use of the second Chern class map. The first goal of the present thesis is to extend this connection through the use of higher Chern class maps. Our main technical tool is the Steinberg basis, which provides explicit generators for the γ-filtration on the Grothendieck group K_0(X) in terms of characteristic classes of line bundles over X. As an application, we establish a connection between the J-invariant and the Tits algebras of a group G of inner type E6. The second goal of this thesis is to relate the indices of the Tits algebras of G to nontrivial torsion elements in the γ-filtration on K_0(X). While the Steinberg basis provides an explicit set of generators of the γ-filtration, the relations are not easily computed. A tool introduced by Zainoulline called the twisted γ-filtration acts as a surjective image of the γ-filtration, with explicit sets of both generators and relations. We use this tool to construct torsion elements in the degree 2 component of the γ-filtration for groups of inner type D2n. Such a group corresponds to an algebra A endowed with an orthogonal involution having trivial discriminant. In the trialitarian case (i.e. type D4), we construct a specific element in the γ-filtration which detects splitting of the associated Tits algebras. We then relate the non-triviality of this element to other properties of the trialitarian triple such as decomposability and hyperbolicity.
- Published
- 2014
- Full Text
- View/download PDF
17. The lattice automorphisms of simple algebraic groups over $$\bar F_2 $$
- Author
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Mauro Costantini
- Subjects
Weyl group ,Lattice automorphism ,General Mathematics ,Algebraic geometry ,ALGEBRAIC GROUPS ,Unipotent ,Reductive group ,Automorphism ,Combinatorics ,symbols.namesake ,Conjugacy class ,Algebraic group ,symbols ,Algebraic number ,Mathematics - Abstract
We prove that if u is a unipotent element of a connected reductive algebraic group G over\(\bar F_2 \), there exists an involution σ in G such that σuσ=u−1. We use this result to determine the group of lattice automorphisms of G, when G is simple.
- Published
- 1996
18. Quantum automata and algebraic groups
- Author
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Derksen, Harm, Jeandel, Emmanuel, Koiran, Pascal, Laboratoire de l'Informatique du Parallélisme (LIP), Centre National de la Recherche Scientifique (CNRS)-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École normale supérieure - Lyon (ENS Lyon), Laboratoire de l'informatique du parallélisme, École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL)
- Subjects
Automates probabilistes ,Algebraic Groups ,Automates quantiques ,Géométrie algébrique ,Quantum Automata ,Indécidabilité ,[INFO]Computer Science [cs] ,Groupes algébriques ,Algebraic Geometry ,Probabilistic Automata ,Undecidability - Abstract
We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices.; Nous montrons ici que plusieurs problèmes indécidables pour des automates probabilistes sont décidables pour des automates quantiques.Ce résultat s’appuie sur un algorithme intéressant en soi, qui, étant donné des matrices inversibles, calcule la clôture de Zariski du groupe engendré par ses matrices.
- Published
- 2003
19. Rational Points of Commutator Subgroups of Solvable Algebraic Groups
- Author
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Fauntleroy, Amassa
- Published
- 1974
- Full Text
- View/download PDF
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