Let U(q) be a Sylow p-subgroup of the Chevalley groups D4(q) where q is a power of a prime p. We ... more Let U(q) be a Sylow p-subgroup of the Chevalley groups D4(q) where q is a power of a prime p. We describe a construction of all complex irreducible characters of U(q) and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters. Furthermore, we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in (q−1) with nonnegative integer coefficients.
We develop theorems which produce a multitude of hyperbolic triples for the finite classical grou... more We develop theorems which produce a multitude of hyperbolic triples for the finite classical groups. We apply these theorems to prove that every quasisimple group except Alt(5) and SL_2(5) is a Beauville group. In particular, we settle a conjecture of Bauer, Catanese and Grunewald which asserts that all non-abelian finite simple groups except for the alternating group Alt(5) are Beauville groups.
Let q be a power of a prime p, let G be a finite Chevalley group over F q and let U be a Sylow p-... more Let q be a power of a prime p, let G be a finite Chevalley group over F q and let U be a Sylow p-subgroup of G; we assume that p is not a very bad prime for G. We explain a procedure of reduction of irreducible complex characters of U , which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of U along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when G is of type F 4 , where we observe that the parametrization is "uniform" over good primes p > 3, but differs for the bad prime p = 3. We also explain how it has been applied for all groups of rank 4 or less.
We present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over ... more We present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over a finite field. This is a necessary component for some constructive recognition algorithms of exceptional quasisimple groups of Lie type. When applied to a simple Chevalley Lie algebra in characteristic p ≥ 5, our algorithm has complexity involving the 7th power of the Lie rank, which is likely to be close to best possible.
We describe a new algorithm for computing braid orbits on Nielsen classes. As an application we c... more We describe a new algorithm for computing braid orbits on Nielsen classes. As an application we classify all families of affine genus zero systems; that is all families of coverings of the Riemann sphere by itself such that the monodromy group is a primitive affine permutation group.
We determine the image of the braid groups inside the Iwahori-Hecke algebras of type A, when defi... more We determine the image of the braid groups inside the Iwahori-Hecke algebras of type A, when defined over a finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter.
We provide estimates for the xed point ratios in the permutation representations of a nite classi... more We provide estimates for the xed point ratios in the permutation representations of a nite classical group over a eld of order q on k-subspaces of its natural n-dimensional module. For su ciently large n each element must either have a negligible ratio or act linearly with a large eigenspace. We obtain an asymptotic estimate in the latter case, which in most cases is q ?dk where d is the codimension of the large eigenspace. We also obtain more speci c results that are applicable to the monodromy conjecture of Guralnick and Thompson. L to the permutation action of Gal(N=C (t)), and the work of several authors shows that to establish the conjecture it su ces to consider primitive permutation actions of nearly simple groups. (See 8] for a more detailed discussion.)
Transactions of the American Mathematical Society, 2013
If a black box group is known to be isomorphic to an exceptional simple group of Lie type of (twi... more If a black box group is known to be isomorphic to an exceptional simple group of Lie type of (twisted) rank > 1, other than any 2 F 4 (q), over a field of known size, a Las Vegas algorithm is given to produce a constructive isomorphism. In view of its timing, this algorithm yields an upgrade of all known nearly linear time Monte Carlo permutation group algorithms to Las Vegas algorithms when the input group has no composition factor isomorphic to any group 2 F 4 (q) or 2 G 2 (q).
Let C_g be a general curve of genus g>3. Guralnick and others proved that the monodromy group of ... more Let C_g be a general curve of genus g>3. Guralnick and others proved that the monodromy group of a cover C_g-> P^1 of degree n is either S_n or A_n. We show that A_n occurs for n>2g. The corresponding result for S_n is classical.
ABSTRACT In this paper we analyze the structure of transitive permutation groups that have trivia... more ABSTRACT In this paper we analyze the structure of transitive permutation groups that have trivial four point stabilizers, but some nontrivial three point stabilizer. In particular we give a complete, detailed classification when the group is simple or quasisimple. This paper is motivated by questions concerning the relationship between fixed points of automorphisms of Riemann surfaces and Weierstrass points and is a continuation of the authors' earlier work.
Let U(q) be a Sylow p-subgroup of the Chevalley groups D4(q) where q is a power of a prime p. We ... more Let U(q) be a Sylow p-subgroup of the Chevalley groups D4(q) where q is a power of a prime p. We describe a construction of all complex irreducible characters of U(q) and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters. Furthermore, we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in (q−1) with nonnegative integer coefficients.
We develop theorems which produce a multitude of hyperbolic triples for the finite classical grou... more We develop theorems which produce a multitude of hyperbolic triples for the finite classical groups. We apply these theorems to prove that every quasisimple group except Alt(5) and SL_2(5) is a Beauville group. In particular, we settle a conjecture of Bauer, Catanese and Grunewald which asserts that all non-abelian finite simple groups except for the alternating group Alt(5) are Beauville groups.
Let q be a power of a prime p, let G be a finite Chevalley group over F q and let U be a Sylow p-... more Let q be a power of a prime p, let G be a finite Chevalley group over F q and let U be a Sylow p-subgroup of G; we assume that p is not a very bad prime for G. We explain a procedure of reduction of irreducible complex characters of U , which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of U along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when G is of type F 4 , where we observe that the parametrization is "uniform" over good primes p > 3, but differs for the bad prime p = 3. We also explain how it has been applied for all groups of rank 4 or less.
We present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over ... more We present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over a finite field. This is a necessary component for some constructive recognition algorithms of exceptional quasisimple groups of Lie type. When applied to a simple Chevalley Lie algebra in characteristic p ≥ 5, our algorithm has complexity involving the 7th power of the Lie rank, which is likely to be close to best possible.
We describe a new algorithm for computing braid orbits on Nielsen classes. As an application we c... more We describe a new algorithm for computing braid orbits on Nielsen classes. As an application we classify all families of affine genus zero systems; that is all families of coverings of the Riemann sphere by itself such that the monodromy group is a primitive affine permutation group.
We determine the image of the braid groups inside the Iwahori-Hecke algebras of type A, when defi... more We determine the image of the braid groups inside the Iwahori-Hecke algebras of type A, when defined over a finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter.
We provide estimates for the xed point ratios in the permutation representations of a nite classi... more We provide estimates for the xed point ratios in the permutation representations of a nite classical group over a eld of order q on k-subspaces of its natural n-dimensional module. For su ciently large n each element must either have a negligible ratio or act linearly with a large eigenspace. We obtain an asymptotic estimate in the latter case, which in most cases is q ?dk where d is the codimension of the large eigenspace. We also obtain more speci c results that are applicable to the monodromy conjecture of Guralnick and Thompson. L to the permutation action of Gal(N=C (t)), and the work of several authors shows that to establish the conjecture it su ces to consider primitive permutation actions of nearly simple groups. (See 8] for a more detailed discussion.)
Transactions of the American Mathematical Society, 2013
If a black box group is known to be isomorphic to an exceptional simple group of Lie type of (twi... more If a black box group is known to be isomorphic to an exceptional simple group of Lie type of (twisted) rank > 1, other than any 2 F 4 (q), over a field of known size, a Las Vegas algorithm is given to produce a constructive isomorphism. In view of its timing, this algorithm yields an upgrade of all known nearly linear time Monte Carlo permutation group algorithms to Las Vegas algorithms when the input group has no composition factor isomorphic to any group 2 F 4 (q) or 2 G 2 (q).
Let C_g be a general curve of genus g>3. Guralnick and others proved that the monodromy group of ... more Let C_g be a general curve of genus g>3. Guralnick and others proved that the monodromy group of a cover C_g-> P^1 of degree n is either S_n or A_n. We show that A_n occurs for n>2g. The corresponding result for S_n is classical.
ABSTRACT In this paper we analyze the structure of transitive permutation groups that have trivia... more ABSTRACT In this paper we analyze the structure of transitive permutation groups that have trivial four point stabilizers, but some nontrivial three point stabilizer. In particular we give a complete, detailed classification when the group is simple or quasisimple. This paper is motivated by questions concerning the relationship between fixed points of automorphisms of Riemann surfaces and Weierstrass points and is a continuation of the authors' earlier work.
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Papers by Kay Magaard