Decomposition numbers modulo p of certain representations of the groups $SL_n(p^k), SU_n(p^k), Sp_{2n}(p^k)$

A. Zalesskii

Displaying similar documents to “Decomposition numbers modulo p of certain representations of the groups $S L_{n} (p^{k}), S U_{n} (p^{k}), S p_{2 n} (p^{k})$ ”

Representations of modules over a *-algebra and related seminorms

Camillo Trapani, Salvatore Triolo (2008)

Studia Mathematica

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Representations of a module over a *-algebra $_{}$ are considered and some related seminorms are constructed and studied, with the aim of finding bounded *-representations of $_{}$ .

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...

Representations of $S_{n}$ and $G L_{n}$ and the q-Schur algebra

Gordon James (1990)

Banach Center Publications

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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.

On symmetric functions and the spin characters of $S_{n}$

John R. Stembridge (1990)

Banach Center Publications

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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). ...

Integral representations of risk functions for basket derivatives

Michał Barski (2012)

Applicationes Mathematicae

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The risk minimizing problem $E [l ((H - X_{T}^{x, π}) ⁺)] \overset{π}{\to} m i n$ 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.

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 $ℓ$ 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 $ℓ \neq p$ be two different prime numbers, let $F$ be a local non archimedean field of residual characteristic $p$ , and let ${\bar{𝐐}}_{ℓ}, {\bar{𝐙}}_{ℓ}, {\bar{𝐅}}_{ℓ}$ be an algebraic closure of the field of $ℓ$ -adic numbers $𝐐_{ℓ}$ , the ring of integers of ${\bar{𝐐}}_{ℓ}$ , the residual field of ${\bar{𝐙}}_{ℓ}$ . We proved the existence and the unicity of a Langlands local correspondence over ${\bar{𝐅}}_{ℓ}$ for all $n \geq 2$ , compatible with the reduction modulo $ℓ$ in [V5], without using $L$ and $ϵ$ factors of pairs. We conjecture that the Langlands local correspondence over ${\bar{𝐐}}_{ℓ}$ 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 $\leq 1$ . 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 $\sum_{i = 1}^{a} (x ²_{i} + x_{i} y_{i} + y ²_{i}) + 2 \sum_{j = 1}^{b} (u ²_{j} + u_{j} v_{j} + v ²_{j})$ . 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.

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 $𝒵_{Γ} (s) = \sum_{n = 1}^{\infty} r_{n} (Γ) n^{- s}$ . 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 $G = G_{p} \times B$ 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...

Asymptotic representations of solutions of the nonautonomous ordinary differential $n$ -th order equations

Mousa Jaber Abu Elshour, Vjacheslav Evtukhov (2017)

Archivum Mathematicum

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Asymptotic representations of some classes of solutions of nonautonomous ordinary differential $n$ -th order equations which somewhat are close to linear equations are established.

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 $l$ .

О понятиях $M$ - и $D$ -индуцированных представлений.

А.Н. Куземчиков, А.В. Левичев (1995)

Sibirskij matematiceskij zurnal

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Holonomy groups of flat manifolds with the $R_{\infty}$ property

Rafał Lutowski, Andrzej Szczepański (2013)

Fundamenta Mathematicae

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Let M be a flat manifold. We say that M has the $R_{\infty}$ property if the Reidemeister number R(f) is infinite for every homeomorphism f: M → M. We investigate relations between the holonomy representation ρ of M and the $R_{\infty}$ property. When the holonomy group of M is solvable we show that if ρ has a unique ℝ-irreducible subrepresentation of odd degree then M has the $R_{\infty}$ property. This result is related to Conjecture 4.8 in [K. Dekimpe et al., Topol. Methods Nonlinear Anal. 34 (2009)].

On twisted group algebras of OTP representation type

Leonid F. Barannyk, Dariusz Klein (2012)

Colloquium Mathematicae

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Assume that S is a commutative complete discrete valuation domain of characteristic p, S* is the unit group of S and $G = G_{p} \times B$ is 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*). We give necessary and sufficient conditions for $S^{λ} G$ to be of OTP representation type, in the sense that every indecomposable $S^{λ} G$ -module is isomorphic to the outer tensor product V W of an indecomposable $S^{λ} G_{p}$ -module V and an irreducible...

Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations

Daniel Simson (2007)

Colloquium Mathematicae

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We prove that the study of the category C-Comod of left comodules over a K-coalgebra C reduces to the study of K-linear representations of a quiver with relations if K is an algebraically closed field, and to the study of K-linear representations of a K-species with relations if K is a perfect field. Given a field K and a quiver Q = (Q₀,Q₁), we show that any subcoalgebra C of the path K-coalgebra K◻Q containing $K^{◻} Q ₀ \oplus K^{◻} Q ₁$ is the path coalgebra $K^{◻} (Q,)$ of a profinite bound quiver (Q,), and the category...

The expansion of $\prod_{k = 1}^{\infty} (1 - q^{a k}) (1 - q^{b k})$

Zhi-Hong Sun (2008)

Acta Arithmetica

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For Coxeter groups $z^{| g |}$ is a coefficient of a uniformly bounded representation

Tadeusz Januszkiewicz (2002)

Fundamenta Mathematicae

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We prove the theorem in the title by constructing an action of a Coxeter group on a product of trees.

A descriptive view of unitary group representations

Simon Thomas (2015)

Journal of the European Mathematical Society

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In this paper, we will study the relative complexity of the unitary duals of countable groups. In particular, we will explain that if $G$ and $H$ are countable amenable non-type I groups, then the unitary duals of $G$ and $H$ are Borel isomorphic.

Entropy of Schur–Weyl measures

Sevak Mkrtchyan (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Relative dimensions of isotypic components of $N$ th order tensor representations of the symmetric group on $n$ letters give a Plancherel-type measure on the space of Young diagrams with $n$ cells and at most $N$ rows. It was conjectured by G. Olshanski that dimensions of isotypic components of tensor representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to this family of Plancherel-type measures in the limit when $\frac{N}{\sqrt{n}}$ converges to a constant....

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 ) ”

Displaying similar documents to “Decomposition numbers modulo p of certain representations of the groups $S L_{n} (p^{k}), S U_{n} (p^{k}), S p_{2 n} (p^{k})$ ”