6 results
Search Results
2. On the characters of the Sylow -subgroups of untwisted Chevalley groups
- Author
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Tung Le, Frank Himstedt, and Kay Magaard
- Subjects
Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Sylow theorems ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Prime (order theory) ,Combinatorics ,Character (mathematics) ,Computational Theory and Mathematics ,Group of Lie type ,Rank (graph theory) ,0101 mathematics ,Subquotient ,Mathematics - Abstract
Let$UY_{n}(q)$be a Sylow$p$-subgroup of an untwisted Chevalley group$Y_{n}(q)$of rank$n$defined over $\mathbb{F}_{q}$where$q$is a power of a prime$p$. We partition the set$\text{Irr}(UY_{n}(q))$of irreducible characters of$UY_{n}(q)$into families indexed by antichains of positive roots of the root system of type$Y_{n}$. We focus our attention on the families of characters of$UY_{n}(q)$which are indexed by antichains of length$1$. Then for each positive root$\unicode[STIX]{x1D6FC}$we establish a one-to-one correspondence between the minimal degree members of the family indexed by$\unicode[STIX]{x1D6FC}$and the linear characters of a certain subquotient$\overline{T}_{\unicode[STIX]{x1D6FC}}$of$UY_{n}(q)$. For$Y_{n}=A_{n}$our single root character construction recovers, among other things, the elementary supercharacters of these groups. Most importantly, though, this paper lays the groundwork for our classification of the elements of$\text{Irr}(UE_{i}(q))$,$6\leqslant i\leqslant 8$, and$\text{Irr}(UF_{4}(q))$.
- Published
- 2016
3. Finding 47:23 in the Baby Monster
- Author
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Robert A. Wilson, John N. Bray, and Richard A Parker
- Subjects
010101 applied mathematics ,Combinatorics ,Computational Theory and Mathematics ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Test (assessment) ,Monster ,Mathematics - Abstract
In this paper we describe methods for finding very small maximal subgroups of very large groups, with particular application to the subgroup 47:23 of the Baby Monster. This example is completely intractable by standard or naïve methods. The example of finding 31:15 inside the Thompson group $\text{Th}$ is also discussed as a test case.
- Published
- 2016
4. Fast heuristic algorithms for computing relations in the class group of a quadratic order, with applications to isogeny evaluation
- Author
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Jean-François Biasse, Michael J. Jacobson, and Claus Fieker
- Subjects
Isogeny ,Discrete mathematics ,Ideal (set theory) ,Heuristic ,General Mathematics ,010102 general mathematics ,Ideal class group ,02 engineering and technology ,01 natural sciences ,Prime (order theory) ,Algebra ,Finite field ,Computational Theory and Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Order (group theory) ,020201 artificial intelligence & image processing ,Quadratic field ,0101 mathematics ,Algorithm ,Mathematics - Abstract
In this paper, we present novel algorithms for finding small relations and ideal factorizations in the ideal class group of an order in an imaginary quadratic field, where both the norms of the prime ideals and the size of the coefficients involved are bounded. We show how our methods can be used to improve the computation of large-degree isogenies and endomorphism rings of elliptic curves defined over finite fields. For these problems, we obtain improved heuristic complexity results in almost all cases and significantly improved performance in practice. The speed-up is especially high in situations where the ideal class group can be computed in advance.
- Published
- 2016
5. On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients
- Author
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Alina Bucur, Almasa Odžak, Anne-Maria Ernvall-Hytönen, and Lejla Smajlović
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Dirichlet L-function ,010103 numerical & computational mathematics ,Dirichlet eta function ,01 natural sciences ,Riemann zeta function ,Riemann hypothesis ,symbols.namesake ,Dirichlet kernel ,Computational Theory and Mathematics ,Dirichlet's principle ,symbols ,0101 mathematics ,Selberg class ,Dirichlet series ,Mathematics - Abstract
The Li coefficients $\unicode[STIX]{x1D706}_{F}(n)$ of a zeta or $L$-function $F$ provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the $\unicode[STIX]{x1D70F}$-Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport–Heilbronn zeta function. The behavior of the $\unicode[STIX]{x1D70F}$-Li coefficients varies depending on whether the function in question has any zeros in the half-plane $\text{Re}(z)>\unicode[STIX]{x1D70F}/2.$ We investigate analytically and numerically the behavior of these coefficients for such functions in both the $n$ and $\unicode[STIX]{x1D70F}$ aspects.
- Published
- 2016
6. Gelfand–Kirillov dimension of differential difference algebras
- Author
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Xiangui Zhao and Yang Zhang
- Subjects
Pure mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Differential difference equations ,Mathematics - Rings and Algebras ,0102 computer and information sciences ,General Medicine ,010103 numerical & computational mathematics ,Difference algebra ,01 natural sciences ,Upper and lower bounds ,Algebra ,Computational Theory and Mathematics ,Dimension (vector space) ,Rings and Algebras (math.RA) ,010201 computation theory & mathematics ,16P90, 16S36 ,Gelfand–Kirillov dimension ,FOS: Mathematics ,0101 mathematics ,Mathematics::Representation Theory ,Differential (mathematics) ,Mathematics - Abstract
Differential difference algebras were introduced by Mansfield and Szanto, which arose naturally from differential difference equations. In this paper, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We give a lower bound of the Gelfand-Kirillov dimension of a differential difference algebra and a sufficient condition under which the lower bound is reached; we also find an upper bound of this Gelfand-Kirillov dimension under some specific conditions and construct an example to show that this upper bound can not be sharpened any more., Comment: 12 pages
- Published
- 2014
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