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Sep 24, 2013
09/13

by
Robert Guralnick; Pham Huu Tiep

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We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we characterize solvable groups and more generally p-solvable groups in terms of containing a triple of elements of distinct prime power orders with product 1. This is also related to various questions about Fried's modular tower program and properties of Hurwitz spaces of covers of curves.

Source: http://arxiv.org/abs/1112.4559v1

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40

Sep 19, 2013
09/13

by
Robert Guralnick; Pham Huu Tiep

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The notion of age of elements of complex linear groups was introduced by M. Reid and is of importance in algebraic geometry, in particular in the study of crepant resolutions and of quotients of Calabi-Yau varieties. In this paper, we solve a problem raised by J. Kollar and M. Larsen on the structure of finite irreducible linear groups generated by elements of age at most 1. More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation. As...

Source: http://arxiv.org/abs/1009.2535v1

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Sep 21, 2013
09/13

by
Alexander S. Kleshchev; Pham Huu Tiep

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We classify the irreducible projective representations of symmetric and alternating groups of minimal possible and second minimal possible dimensions, and get a lower bound for the third minimal dimension. On the way we obtain some new results on branching which might be of independent interest.

Source: http://arxiv.org/abs/1106.3123v2

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5.0

Jun 29, 2018
06/18

by
Nguyen Ngoc Hung; Pham Huu Tiep

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The classical It\^o-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group $G$ is coprime to a given prime $p$, then $G$ has a normal Sylow $p$-subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of $G$ is less than $4/3$ then $G$ has a normal Sylow...

Topics: Group Theory, Representation Theory, Mathematics

Source: http://arxiv.org/abs/1606.05807

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Sep 22, 2013
09/13

by
Robert M. Guralnick; Pham Huu Tiep

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Let G be a finite quasisimple group of Lie type. We show that there are regular semisimple elements x,y in G, x of prime order, and |y| is divisible by at most two primes, such that the product of the conjugacy classes of x and y contain all non-central elements of G. In fact in all but four cases, y can be chosen to be of square-free order. Using this result, we prove an effective version of one of the main results of Larsen, Shalev and Tiep by showing that, given any positive integer m, if...

Source: http://arxiv.org/abs/1302.0333v1

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Sep 19, 2013
09/13

by
Hung Ngoc Nguyen; Pham Huu Tiep

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We prove that the restriction of any absolutely irreducible representation of Steinberg's triality groups $^3D_4(q)$ in characteristic coprime to q to any proper subgroup is reducible

Source: http://arxiv.org/abs/0910.4888v1

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Sep 22, 2013
09/13

by
Alexander S. Kleshchev; Pham Huu Tiep

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We classify all triples $(G,V,H)$ such that $SL_n(q)\leq G\leq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $\FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $V\dar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups.

Source: http://arxiv.org/abs/0811.2766v1

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Sep 23, 2013
09/13

by
Pham Huu Tiep; Alexander E. Zalesskii

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We prove an analogue of the celebrated Hall-Higman theorem, which gives a lower bound for the degree of the minimal polynomial of any semisimple element of prime power order $p^{a}$ of a finite classical group in any nontrivial irreducible cross characteristic representation. With a few explicit exceptions, this degree is at least $p^{a-1}(p-1)$.

Source: http://arxiv.org/abs/0810.0855v1

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46

Sep 23, 2013
09/13

by
Alexander S. Kleshchev; Pham Huu Tiep

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We determine precisely the number of irreducible summands of an irreducible cross characteristic representation of $GL_{n}(q)$ on restriction to $SL_{n}(q)$. Combined with a recent result of C. Bonnafe, this yields a canonical labeling for irreducible $\ell$-modular representations of $SL_{n}(q)$, where $(\ell,q)=1$. As an application, we classify for the first time complex representations of $SL_{n}(q)$ whose reductions modulo $\ell$ are irreducible.

Source: http://arxiv.org/abs/0810.0849v1

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Sep 23, 2013
09/13

by
Robert M. Guralnick; Pham Huu Tiep

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We prove a conjecture of Kollar and Larsen on Zariski closed subgroups of $GL(V)$ which act irreducibly on some symmetric power $Sym^{k}(V)$ with $k \geq 4$. This conjecture has interesting implications, in particular on the holonomy group of a stable vector bundle on a smooth projective variety, as shown by the recent work of Balaji and Kollar.

Source: http://arxiv.org/abs/0810.0853v1

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Sep 18, 2013
09/13

by
Robert M. Guralnick; Pham Huu Tiep

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We classify all pairs (G,V) with G a closed subgroup in a classical group with natural module V over the complex numbers such that G has the same composition factors on the kth tensor power of V, for a fixed (small) k. In particular, we prove Larsen's conjecture stating that for dim(V) > 6 and k = 4, there are no such G aside from those containing the derived subgroup of the classical group. We also find all the examples where this fails for dim(V) < 7. As a consequence of our results, we...

Source: http://arxiv.org/abs/math/0502080v1

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8.0

Jun 30, 2018
06/18

by
Alexander Kleshchev; Peter Sin; Pham Huu Tiep

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We prove that non-trivial representations of the alternating group $A_n$ are reducible over a primitive proper subgroup which is isomorphic to some alternating group $A_m$.

Topics: Mathematics, Representation Theory, Group Theory

Source: http://arxiv.org/abs/1405.3324

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6.0

Jun 30, 2018
06/18

by
Robert Guralnick; Florian Herzig; Pham Huu Tiep

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The notion of adequate subgroups was introduced by Jack Thorne [59]. It is a weakening of the notion of big subgroups used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] and [23] that if the dimension is smaller than the characteristic then almost all absolutely irreducible representations are adequate. We extend the results by considering all...

Topics: Mathematics, Number Theory, Representation Theory, Group Theory

Source: http://arxiv.org/abs/1405.0043

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Sep 19, 2013
09/13

by
Michael Larsen; Gunter Malle; Pham Huu Tiep

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Answering a question of I. M. Isaacs, we show that the largest degree of irreducible complex representations of any finite non-abelian simple group can be bounded in terms of the smaller degrees. We also study the asymptotic behavior of this largest degree for finite groups of Lie type. Moreover, we show that for groups of Lie type, the Steinberg character has largest degree among all unipotent characters.

Source: http://arxiv.org/abs/1010.5628v1

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22

Jun 28, 2018
06/18

by
Karen Meagher; Pablo Spiga; Pham Huu Tiep

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We prove an analogue of the classical Erd\H{o}s-Ko-Rado theorem for intersecting sets of permutations in finite 2-transitive groups. Given a finite group G acting faithfully and 2-transitively on the set X, we show that an intersecting set of maximal size in G has cardinality |G|/|X|. This generalises and gives a unifying proof of some similar recent results in the literature.

Topics: Group Theory, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1507.06450

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Sep 22, 2013
09/13

by
Jason Fulman; Jan Saxl; Pham Huu Tiep

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We develop cycle index generating functions for orthogonal groups in even characteristic, and give some enumerative applications. A key step is the determination of the values of the complex linear-Weil characters of the finite symplectic group, and their inductions to the general linear group, at unipotent elements. We also define and study several natural probability measures on integer partitions.

Source: http://arxiv.org/abs/1004.2678v1

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Sep 19, 2013
09/13

by
Robert Guralnick; Michael Larsen; Pham Huu Tiep

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We study the representation growth of alternating and symmetric groups in positive characteristic and restricted representation growth for the finite groups of Lie type. We show that the the number of representations of dimension at most n is bounded by a low degree polynomial in n. As a consequence, we show that the number of conjugacy classes of maximal subgroups of a finite almost simple group G is at most O(log|G|).

Source: http://arxiv.org/abs/1009.2437v2

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Jul 20, 2013
07/13

by
Michael Larsen; Aner Shalev; Pham Huu Tiep

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The classical Waring problem deals with expressing every natural number as a sum of g(k) kth powers. Similar problems for finite simple groups have been studied recently, and in this paper we study them for finite quasisimple groups G. We show that for a fixed non-trivial group word w and large enough G we have w(G)^3=G, namely every element of G is a product of 3 values of w. For various families of finite quasisimple groups, including covers of alternating groups, we obtain a stronger result,...

Source: http://arxiv.org/abs/1107.3341v1

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6.0

Jun 29, 2018
06/18

by
Robert M. Guralnick; Gabriel Navarro; Pham Huu Tiep

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We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight conjecture for these groups.

Topics: Group Theory, Representation Theory, Mathematics

Source: http://arxiv.org/abs/1602.04797

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Sep 19, 2013
09/13

by
Frank Himstedt; Hung Ngoc Nguyen; Pham Huu Tiep

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We prove that the restriction of any nontrivial representation of the Ree groups $^2F_{4}(q), q=2^{2n+1}\geq8$ in odd characteristic to any proper subgroup is reducible. We also determine all triples $(K, V, H)$ such that $K \in \{^2F_4(2), ^2F_4(2)'\}$, $H$ is a proper subgroup of $K$, and $V$ is a representation of $K$ in odd characteristic restricting absolutely irreducibly to $H$.

Source: http://arxiv.org/abs/0910.4758v1

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51

Sep 23, 2013
09/13

by
Robert M. Guralnick; Gunter Malle; Pham Huu Tiep

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We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two...

Source: http://arxiv.org/abs/1202.2627v3

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6.0

Jun 29, 2018
06/18

by
Eugenio Giannelli; Alexander Kleshchev; Gabriel Navarro; Pham Huu Tiep

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Let $q$ be an odd prime power, $n > 1$, and let $P$ denote a maximal parabolic subgroup of $GL_n(q)$ with Levi subgroup $GL_{n-1}(q) \times GL_1(q)$. We restrict the odd-degree irreducible characters of $GL_n(q)$ to $P$ to discover a natural correspondence of characters, both for $GL_n(q)$ and $SL_n(q)$. A similar result is established for certain finite groups with self-normalizing Sylow $p$-subgroups. We also construct a canonical bijection between the odd-degree irreducible characters of...

Topics: Representation Theory, Group Theory, Mathematics

Source: http://arxiv.org/abs/1601.04423

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Sep 20, 2013
09/13

by
P. Oscar Boykin; Meera Sitharam; Pham Huu Tiep; Pawel Wocjan

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We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form, where K=R or K=C. In particular, a complete collection of MUBs in C^n gives rise to a so-called orthogonal...

Source: http://arxiv.org/abs/quant-ph/0506089v1

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Sep 18, 2013
09/13

by
Gerhard Heide; Jan Saxl; Pham Huu Tiep; Alexandre E. Zalesski

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Let $G$ be a finite simple group of Lie type, and let $\pi_G$ be the permutation representation of $G$ associated with the action of $G$ on itself by conjugation. We prove that every irreducible representation of $G$ is a constituent of $\pi_G$, unless $G=PSU_n(q)$ and $n$ is coprime to $2(q+1)$, where precisely one irreducible representation fails. Let St be the Steinberg representation of $G$. We prove that a complex irreducible representation of $G$ is a constituent of the tensor square...

Source: http://arxiv.org/abs/1209.1768v1