Displaying similar documents to “Decomposition numbers modulo p of certain representations of the groups S L n ( p k ) , S U n ( p k ) , S p 2 n ( p k )

On the representation theory of braid groups

Ivan Marin (2013)

Annales mathématiques Blaise Pascal

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This work presents an approach towards the representation theory of the braid groups B n . We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids, with the help of Drinfeld associators. We set a dictionary between representation-theoretic properties of these two structures, and tools to describe the representations thus obtained. We give an explanation for the frequent apparition of unitary structures...

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 [ X ] [ 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.

Stacks of group representations

Paul Balmer (2015)

Journal of the European Mathematical Society

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We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group G , the derived and the stable categories of representations of a subgroup H can be constructed out of the corresponding category for G by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods...

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). ...

A note on representing dowling geometries by partitions

František Matúš, Aner Ben-Efraim (2020)

Kybernetika

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We prove that a rank 3 Dowling geometry of a group H is partition representable if and only if H is a Frobenius complement. This implies that Dowling group geometries are secret-sharing if and only if they are multilinearly representable.

Integral representations of risk functions for basket derivatives

Michał Barski (2012)

Applicationes Mathematicae

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The risk minimizing problem E [ l ( ( H - X T x , π ) ) ] π m i n in the multidimensional Black-Scholes framework is studied. Specific formulas for the minimal risk function and the cost reduction function for basket derivatives are shown. Explicit integral representations for the risk functions for l(x) = x and l ( x ) = x p , with p > 1 for digital, quantos, outperformance and spread options are derived.

Spin representations and binary numbers

Henrik Winther (2024)

Archivum Mathematicum

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We consider a construction of the fundamental spin representations of the simple Lie algebras 𝔰𝔬 ( n ) in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a -graded associative algebra (rather than the usual -filtered Clifford algebra). Our description gives a quick way to write down the spin matrices, and gives a way to encode some extra structure, such as the real structure which is invariant under the compact real form, for some n ....

On indecomposable projective representations of finite groups over fields of characteristic p > 0

Leonid F. Barannyk, Kamila Sobolewska (2003)

Colloquium Mathematicae

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Let G be a finite group, F a field of characteristic p with p||G|, and F λ G the twisted group algebra of the group G and the field F with a 2-cocycle λ ∈ Z²(G,F*). We give necessary and sufficient conditions for F λ G to be of finite representation type. We also introduce the concept of projective F-representation type for the group G (finite, infinite, mixed) and we exhibit finite groups of each type.

Congruences modulo between ϵ factors for cuspidal representations of G L ( 2 )

Marie-France Vignéras (2000)

Journal de théorie des nombres de Bordeaux

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Let p be two different prime numbers, let F be a local non archimedean field of residual characteristic p , and let 𝐐 ¯ , 𝐙 ¯ , 𝐅 ¯ be an algebraic closure of the field of -adic numbers 𝐐 , the ring of integers of 𝐐 ¯ , the residual field of 𝐙 ¯ . We proved the existence and the unicity of a Langlands local correspondence over 𝐅 ¯ for all n 2 , compatible with the reduction modulo in [V5], without using L and ϵ factors of pairs. We conjecture that the Langlands local correspondence over 𝐐 ¯ respects congruences...

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 1 . More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation....

On the number of representations of a positive integer by certain quadratic forms

Ernest X. W. Xia (2014)

Colloquium Mathematicae

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For natural numbers a,b and positive integer n, let R(a,b;n) denote the number of representations of n in the form i = 1 a ( x ² i + x i y i + y ² i ) + 2 j = 1 b ( u ² j + u j v j + v ² j ) . Lomadze discovered a formula for R(6,0;n). Explicit formulas for R(1,5;n), R(2,4;n), R(3,3;n), R(4,2;n) and R(5,1;n) are determined in this paper by using the (p;k)-parametrization of theta functions due to Alaca, Alaca and Williams.

Representations of a class of positively based algebras

Shiyu Lin, Shilin Yang (2023)

Czechoslovak Mathematical Journal

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We investigate the representation theory of the positively based algebra A m , d , which is a generalization of the noncommutative Green algebra of weak Hopf algebra corresponding to the generalized Taft algebra. It turns out that A m , d is of finite representative type if d 4 , of tame type if d = 5 , and of wild type if d 6 . In the case when d 4 , all indecomposable representations of A m , d are constructed. Furthermore, their right cell representations as well as left cell representations of A m , d are described. ...

Representation growth of linear groups

Michael Larsen, Alexander Lubotzky (2008)

Journal of the European Mathematical Society

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Let Γ be a group and r n ( Γ ) the number of its n -dimensional irreducible complex representations. We define and study the associated representation zeta function 𝒵 Γ ( s ) = n = 1 r n ( Γ ) n - s . When Γ is an arithmetic group satisfying the congruence subgroup property then 𝒵 Γ ( s ) has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite...

Tempered reductive homogeneous spaces

Yves Benoist, Toshiyuki Kobayashi (2015)

Journal of the European Mathematical Society

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Let G be a semisimple algebraic Lie group and H a reductive subgroup. We find geometrically the best even integer p for which the representation of G in L 2 ( G / H ) is almost L p . As an application, we give a criterion which detects whether this representation is tempered.

Finite groups of OTP projective representation type over a complete discrete valuation domain of positive characteristic

Leonid F. Barannyk, Dariusz Klein (2012)

Colloquium Mathematicae

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Let S be a commutative complete discrete valuation domain of positive characteristic p, S* the unit group of S, Ω a subgroup of S* and G = G p × B a finite group, where G p is a p-group and B is a p’-group. Denote by S λ G the twisted group algebra of G over S with a 2-cocycle λ ∈ Z²(G,S*). For Ω satisfying a specific condition, we give necessary and sufficient conditions for G to be of OTP projective (S,Ω)-representation type, in the sense that there exists a cocycle λ ∈ Z²(G,Ω) such that every indecomposable...

Irreducibility of automorphic Galois representations of G L ( n ) , n at most 5

Frank Calegari, Toby Gee (2013)

Annales de l’institut Fourier

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Let π be a regular, algebraic, essentially self-dual cuspidal automorphic representation of GL n ( 𝔸 F ) , where F is a totally real field and n is at most 5 . We show that for all primes l , the l -adic Galois representations associated to π are irreducible, and for all but finitely many primes l , the mod l Galois representations associated to π are also irreducible. We also show that the Lie algebras of the Zariski closures of the l -adic representations are independent of l .