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We classify the maximal algebraic subgroups of Bir(C x ℙ¹), when C is a smooth CM projective curve of positive genus. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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.

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]

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]

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]

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]

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]

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]