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382
The inverse Galois problem and rational points on moduli spaces
 Math. Annalen
, 1991
"... Abstract: We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P(x) with L regular over any PAC field P of characteri ..."
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Cited by 56 (26 self)
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Abstract: We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P(x) with L regular over any PAC field P of characteristic zero. A special case of this implies that G is a Galois group over Fp(x) for almost all primes p. Many attempts have been made to realize finite groups as Galois groups of extensions of Q(x) that are regular over Q (see the end of this introduction for definitions). We call this the “regular inverse Galois problem. ” We show that to each finite group G with trivial center and integer r ≥ 3 there is canonically associated an algebraic variety, Hin r (G), defined over Q (usually reducible) satisfying the following.
Random walk in a Weyl chamber
 Proc. Amer. Math. Soc
, 1992
"... Abstract. The classical Ballot problem that counts the number of ways of walking from the origin and staying within the wedge xx> X2> ■> x„ (which is a Weyl chamber for the symmetric group), using positive unit steps, is generalized to general Weyl groups and general sets of steps. To any s ..."
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Cited by 40 (4 self)
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Abstract. The classical Ballot problem that counts the number of ways of walking from the origin and staying within the wedge xx> X2> ■> x„ (which is a Weyl chamber for the symmetric group), using positive unit steps, is generalized to general Weyl groups and general sets of steps. To any simple and natural proof, one can ask the question: How far can it be generalized? We will attempt to give one possible answer to this question for Andre's [A] celebrated solution of the twocandidate ballot problem. Andre's proof uses a reflection argument, and we will show that it can be naturally generalized
Refined anisotropic Ktypes and supercuspidal representations
 Pacific J. Math
, 1998
"... Let F be a nonarchimedean local field, and G a connected reductive group defined over F. We classify the representations of G(F) that contain any anisotropic unrefined minimal Ktype satisfying a certain tameness condition. We show that these representations are induced from compact (mod center) sub ..."
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Cited by 32 (6 self)
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Let F be a nonarchimedean local field, and G a connected reductive group defined over F. We classify the representations of G(F) that contain any anisotropic unrefined minimal Ktype satisfying a certain tameness condition. We show that these representations are induced from compact (mod center) subgroups, and we construct corresponding refined minimal Ktypes. 0. Introduction. Let G be any connected reductive group defined over a nonarchimedean local field F of residual characteristic p. Under some tameness assumptions on G, we construct families of positivedepth supercuspidal representations of G = G(F). In particular, we classify (§2.7) the representations of G
A Residue Calculus for Root Systems
, 2000
"... Let V be a finite dimensional real vector space on which a root system is given. Consider a meromorphic function ' on V C = V +iV , the singular locus of which is a locally finite union of hyperplanes of the form f 2 V C j h; i = sg, 2 , s 2 R. Assume ' is of suitable decay in the imagin ..."
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Cited by 31 (13 self)
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Let V be a finite dimensional real vector space on which a root system is given. Consider a meromorphic function ' on V C = V +iV , the singular locus of which is a locally finite union of hyperplanes of the form f 2 V C j h; i = sg, 2 , s 2 R. Assume ' is of suitable decay in the imaginary directions, so that integrals of the form '() d make sense for generic 2 V . A residue calculus is developed that allows shifting . This residue calculus can be used to obtain Plancherel and PaleyWiener theorems on semisimple symmetric spaces.
Smalldiameter Cayley graphs for finite simple groups
, 1989
"... Let S be a subset generating a finite group G. The corresponding Cayley graph '§(G, 5) has ..."
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Cited by 30 (7 self)
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Let S be a subset generating a finite group G. The corresponding Cayley graph '§(G, 5) has
On the Diameter of Finite Groups
 SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1990
"... The diameter of a group G with respect to a set S of generators is the maximum over g 2 G of the length of the shortest word in S [ S 1 representing g. This concept arises in the contexts of efficient communication networks and Rubik's cube type puzzles. "Best" generators (giving mini ..."
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Cited by 27 (4 self)
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The diameter of a group G with respect to a set S of generators is the maximum over g 2 G of the length of the shortest word in S [ S 1 representing g. This concept arises in the contexts of efficient communication networks and Rubik's cube type puzzles. "Best" generators (giving minimum diameter while keeping the number of generators limited) are pertinent to networks, "worst" and "average" generators seem a more adequate model for puzzles. We survey a substantial body of recent work by the authors on these subjects. Regarding the "best" case, we show that while the structure of the group is essentially irrelevant if S is allowed to exceed (log G) 1+c (c > 0), it plays a heavy role when jSj = O(1). In particular, every nonabelian nite simple group has a set of 7 generators giving logarithmic diameter. This cannot happen for groups with an abelian subgroup of bounded index. { Regarding the worst case, we are concerned primarily with permutation groups of degree n and obtain a tight exp((n ln n) 1=2 (1 + o(1))) upper bound. In the average case, the upper bound improves to exp((ln n) 2 (1 + o(1))). As a rst step toward extending this result to simple groups other than An , we establish that almost every pair of elements of a classical simple group G generates G, a result previously proved by J. Dixon for An . In the limited space of this article, we try to illuminate some of the basic underlying techniques.