Symmetry classes of tensors associated with the semi-dihedral groups $SD_{8n}$

Mahdi Hormozi; Kijti Rodtes

Displaying similar documents to “Symmetry classes of tensors associated with the semi-dihedral groups $S D_{8 n}$ ”

Products of conjugacy classes in finite unitary groups $G U (3, q^{2})$ and $S U (3, q^{2})$

S.Yu. Orevkov (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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For the groups $G U (3)$ , $S U (3)$ , $G L (3)$ , $S U (3)$ over a finite field we solve the class product problem, i.e., we give a complete list of $m$ -tuples of conjugacy classes whose product does not contain the identity matrix.

Deformation theory and finite simple quotients of triangle groups I

Michael Larsen, Alexander Lubotzky, Claude Marion (2014)

Journal of the European Mathematical Society

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Let $2 \leq a \leq b \leq c \in ℕ$ with $μ = 1 / a + 1 / b + 1 / c < 1$ and let $T = T_{a, b, c} = 〈 x, y, z : x^{a} = y^{b} = z^{c} = x y z = 1 〉$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$ ? (Classically, for $(a, b, c) = (2, 3, 7)$ and more recently also for general $(a, b, c)$ .) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially...

The density of representation degrees

Martin Liebeck, Dan Segal, Aner Shalev (2012)

Journal of the European Mathematical Society

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For a group $G$ and a positive real number $x$ , define $d_{G} (x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$ . We study the asymptotics of $d_{G} (x)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $α > 0$ such that $d_{G} (x) > x^{α}$ for all large $x$ , or $G$ is virtually abelian (in which case $d_{G} (x)$ is bounded). ...

Limits of relatively hyperbolic groups and Lyndon’s completions

Olga Kharlampovich, Alexei Myasnikov (2012)

Journal of the European Mathematical Society

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We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon’s completion $G^{ℤ [t]}$ of the group $G$ , or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{ℤ [t]}$ containing $G$ is universally equivalent to $G$ . Since finitely...

A problem of Kollár and Larsen on finite linear groups and crepant resolutions

Robert Guralnick, Pham Tiep (2012)

Journal of the European Mathematical Society

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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. Kollár and M. Larsen on the structure of finite irreducible linear groups generated by elements of age $\leq 1$ . More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation....

Coxeter group actions on the complement of hyperplanes and special involutions

Giovanni Felder, A. Veselov (2005)

Journal of the European Mathematical Society

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We consider both standard and twisted actions of a (real) Coxeter group $G$ on the complement $ℳ_{G}$ to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in $G$ and give explicit formulae which describe both actions on the total cohomology $H^{*} (ℳ_{G}, 𝒞)$ in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group $S_{n}$ , the...

On the Davenport constant and group algebras

Daniel Smertnig (2010)

Colloquium Mathematicae

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For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S = g ₁ \cdot . . . \cdot g_{l}$ over G such that $(X^{g ₁} - a ₁) \cdot . . . \cdot (X^{g_{l}} - a_{l}) \neq 0 \in K [G]$ for all $a ₁, . . ., a_{l} \in K^{\times}$ . If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for...

Characterization of the alternating groups by their order and one conjugacy class length

Alireza Khalili Asboei, Reza Mohammadyari (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group, and let $N (G)$ be the set of conjugacy class sizes of $G$ . By Thompson’s conjecture, if $L$ is a finite non-abelian simple group, $G$ is a finite group with a trivial center, and $N (G) = N (L)$ , then $L$ and $G$ are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation)....

On the structural theory of ${II}_{1}$ factors of negatively curved groups

Ionut Chifan, Thomas Sinclair (2013)

Annales scientifiques de l'École Normale Supérieure

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Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor $L Γ$ is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that $L Γ$ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in $Sp (n, 1)$ , $n \geq 2$ , are virtually $W^{*}$ -superrigid.

Permutability of centre-by-finite groups

Brunetto Piochi (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $G$ be a group and $m$ be an integer greater than or equal to $2$ . $G$ is said to be $m$ -permutable if every product of $m$ elements can be reordered at least in one way. We prove that, if $G$ has a centre of finite index $z$ , then $G$ is $(1+[z/2])$ -permutable. More bounds are given on the least $m$ such that $G$ is $m$ -permutable.

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

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We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$ , then for every subset $B$ of $G$ with $| B | > | G | / k^{1 / 3}$ we have $B^{3} = G$ . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k \geq 2$ , then $G$ has a...

Obstruction sets and extensions of groups

Francesca Balestrieri (2016)

Acta Arithmetica

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Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X {(_{k})}^{é t, B r} \subset X {(_{k})}^{B r ₁}$ . In the first part, we apply ideas from the proof of $X {(_{k})}^{é t, B r} = X {(_{k})}^{_{k}}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $\subset \subset_{k}$ are such that $\subset E x t (,_{k})$ , then $X {(_{k})}^{} = X {(_{k})}^{}$ . This allows us to conclude, among other things, that $X {(_{k})}^{é t, B r} = X {(_{k})}^{_{k}}$ and $X {(_{k})}^{S o l, B r ₁} = X {(_{k})}^{S o l_{k}}$ .

Quiver varieties and the character ring of general linear groups over finite fields

Emmanuel Letellier (2013)

Journal of the European Mathematical Society

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Given a tuple $(𝒳_{1}, ..., 𝒳_{k})$ of irreducible characters of $G L_{n} (F_{q})$ we define a star-shaped quiver $Γ$ together with a dimension vector $v$ . Assume that $(𝒳_{1}, ..., 𝒳_{k})$ is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product $𝒳_{1} \otimes \dots \otimes 𝒳_{k}$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(Γ, v)$ . The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied,...

Brauer relations in finite groups

Alex Bartel, Tim Dokchitser (2015)

Journal of the European Mathematical Society

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If $G$ is a non-cyclic finite group, non-isomorphic $G$ -sets $X, Y$ may give rise to isomorphic permutation representations $ℂ [X] ≅ ℂ [Y]$ . Equivalently, the map from the Burnside ring to the rational representation ring of $G$ has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave–Bouc classification in the case of $p$ -groups.

Groups of given intermediate word growth

Laurent Bartholdi, Anna Erschler (2014)

Annales de l’institut Fourier

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We show that there exists a finitely generated group of growth $\sim f$ for all functions $f : ℝ_{+} \to ℝ_{+}$ satisfying $f (2 R) \leq f {(R)}^{2} \leq f (η_{+} R)$ for all $R$ large enough and $η_{+} \approx 2.4675$ the positive root of $X^{3} - X^{2} - 2 X - 4$ . Set $α_{-} = log 2 / log η_{+} \approx 0.7674$ ; then all functions that grow uniformly faster than $exp (R^{α_{-}})$ are realizable as the growth of a group. We also give a family of sum-contracting branched groups of growth $\sim exp (R^{α})$ for a dense set of $α \in [α_{-}, 1]$ .

Injectors of fitting classes of $\mathfrak{G}_{1}$ -groups

Federico Menegazzo, Martin L. Newell (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Fitting classes and injectors are discussed in the class of $\mathfrak{G}_{1}$ -groups. A necessary and sufficient condition for the existence of injectors is given; it is also shown that, when this condition holds, the injectors form a unique conjugacy class.

The Roquette category of finite $p$ -groups

Serge Bouc (2015)

Journal of the European Mathematical Society

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Let $p$ be a prime number. This paper introduces the Roquette category $ℛ_{p}$ of finite $p$ -groups, which is an additive tensor category containing all finite $p$ -groups among its objects. In $ℛ_{p}$ , every finite $p$ -group $P$ admits a canonical direct summand $\partial P$ , called the edge of $P$ . Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$ -groups, and the tensor structure of $ℛ_{p}$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors...

Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Zdeněk Dušek (2015)

Czechoslovak Mathematical Journal

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Let $(M, g)$ be a 4-dimensional Einstein Riemannian manifold. At each point $p$ of $M$ , the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor $R$ at $p$ . In this basis, up to standard symmetries and antisymmetries, just $5$ components of the curvature tensor $R$ are nonzero. For the space of constant curvature, the group $O (4)$ acts as a transformation group between ST bases at $T_{p} M$ and for the so-called 2-stein curvature tensors, the group $Sp (1) \subset SO (4)$ acts as a transformation...

Completely bounded lacunary sets for compact non-abelian groups

Kathryn Hare, Parasar Mohanty (2015)

Studia Mathematica

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In this paper, we introduce and study the notion of completely bounded $Λ_{p}$ sets ( $Λ_{p}^{c b}$ for short) for compact, non-abelian groups G. We characterize $Λ_{p}^{c b}$ sets in terms of completely bounded $L^{p} (G)$ multipliers. We prove that when G is an infinite product of special unitary groups of arbitrarily large dimension, there are sets consisting of representations of unbounded degree that are $Λ_{p}$ sets for all p < ∞, but are not $Λ_{p}^{c b}$ for any p ≥ 4. This is done by showing that the space of completely bounded $L^{p} (G)$ ...

Displaying similar documents to “Symmetry classes of tensors associated with the semi-dihedral groups S D 8 n ”

Displaying similar documents to “Symmetry classes of tensors associated with the semi-dihedral groups $S D_{8 n}$ ”