Cross characteristic representations of symplectic and unitary groups
type: Item
instance of: scholarly article
title: Cross characteristic representations of symplectic and unitary groups
zbMATH Open document ID: 1025.20002
DOI: 10.1016/S0021-8693(02)00527-6
author: Kay Magaard
author: Jan Saxl
author: Pham Huu Tiep
author: Robert Guralnick
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: In this very remarkable and elaborate paper the authors investigate low-dimensional representations in positive, non-defining characteristic for the symplectic and unitary groups. Furthermore, they deal with problems concerning minimal polynomials and quadratic modules. Let \(p\) and \(r\) be two different primes and \(q=p^f\) for some positive integer \(f\); furthermore, let \(k\) be an algebraically closed field of characteristic \(r\). Then the main results read as follows. Theorem 2.1. Let \(G=\text{Sp}_{2n}(q)\) with \(n\geq 2\) and \(q\) odd. Let \(V\) be an irreducible \(kG\)-module of dimension less than \((q^n-1)(q^n-q)/(2(q+1))\). Then \(V\) is either the trivial module, or a Weil module of dimension \((q^n\pm 1)/2\). Theorem 2.2. Let \(G=\text{Sp}_{2n}(q)\) with \(n\geq 2\) and \(q\) odd. Let \(V\) be an irreducible \(kG\)-module such that any long root subgroup of \(G\) has at most \((q-1)/2\) nontrivial linear characters on \(V\). Then \(V\) is either trivial or a Weil module. Theorem 2.3. Let \(G=\text{Sp}_{2n}(q)\) with \(n\geq 3\) and \(q\) odd. For a long root subgroup \(Z:=\{x_\gamma(t)\mid t\in\mathbb{F}_q^*\}\) let \(\Omega_1(Z)\) denote the set of all linear characters of \(Z\) of the form \(\lambda_a\colon x_\gamma(t)\to\varepsilon^{\text{tr}_{\mathbb{F}_q/\mathbb{F}_p}(at)}\) where \(a\in\mathbb{F}_q^*\) is a square and \(\varepsilon=\exp(2\pi i/p)\); similarly \(\Omega_2(Z)\) is the set of all \(\lambda_a\) where \(a\in\mathbb{F}_q^*\) is a nonsquare. Let \(V\) be an irreducible \(kG\)-module satisfying at least one of the following conditions: (1) If \(Y=Y_1\times Y_2\) with (distinct) long root subgroups \(Y_1\) and \(Y_2\) of \(G\), then all nontrivial linear characters of \(Y\) on \(V\) are of the form \(\alpha_1\otimes\alpha_2\) where either \(\alpha_j\in\Omega_1(Y_j)\) for \(j\in \{1,2\}\) or \(\alpha_j\in\Omega_2(Y_j)\) for \(j\in\{1,2\}\). (2) For some \(j\in\{2,\dots,n-1\}\) the restriction of \(V\) to a standard subgroup \(\text{Sp}…
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C34
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1868066
zbMATH Keywords: finite symplectic groups
zbMATH Keywords: finite unitary groups
zbMATH Keywords: Weil representations
zbMATH Keywords: cross characteristic representations
zbMATH Keywords: quadratic modules
zbMATH Keywords: low-dimensional representations
zbMATH Keywords: minimal polynomials
reviewed by: Wolfgang Lempken
MaRDI profile type: Publication
cites work: Q3684278
cites work: Q3800275
cites work: Lower bounds for degrees of irreducible Brauer characters of finite general linear groups
cites work: Die Zerlegungsmatrizen der Gruppen \(PSL(2,p^f)\)
cites work: Quadratic action and the 𝒫(G, V)-theorem in arbitrary characteristic
cites work: Q3997911
cites work: Minimum polynomials and lower bounds for eigenvalue multiplicities of prime-power order elements in representations of classical groups
cites work: Lower bounds for the minima of certain symplectic and unitary group lattices
cites work: Q3955560
cites work: The blocks of finite general linear and unitary groups
cites work: Linear groups over \(\text{GF}(2^k)\) generated by a conjugacy class of a fixed point free element of order 3
cites work: Irreducible brauer characters of the 3-dimensional special unitary groups in non-defining characteristic∗
cites work: Q3971682
cites work: Weil representations associated to finite fields
cites work: Even unimodular lattices associated with the Weil representation of the finite symplectic group
cites work: Group Representations and Lattices
cites work: Small representations are completely reducible
cites work: Linear Groups with Orders Having Certain Large Prime Divisors
cites work: Finite Covers, Cohomology and Homogeneous Structures
cites work: Cross characteristic representations of even characteristic symplectic groups
cites work: Regular and Semisimple Blocks of Finite Reductive Groups
cites work: Low-dimensional representations of special unitary groups
cites work: On the quadratic pairs
cites work: Chevalley groups of odd characteristic as quadratic pairs
cites work: Cross characteristic projective representations for some classical groups
cites work: On the Character of Weil's Representation
cites work: Characters of Solvable and Symplectic Groups
cites work: Q4839675
cites work: On the minimal degrees of projective representations of the finite Chevalley groups
cites work: The modulo 2 structure of rank 3 permutation modules for odd characteristic symplectic groups.
cites work: Permutation modules for rank 3 symplectic and orthogonal groups
cites work: SMALLEST DEGREES OF REPRESENTATIONS OF EXCEPTIONAL GROUPS OF LIE TYPE
cites work: Irreducible tensor products of representations of finite quasi-simple groups of Lie type
cites work: A characterization of the spinmodule for \(2\cdot A_ n\)
cites work: Q5679022
cites work: Q4097918
cites work: Quadratic modules for CHEVALLEY groups over fields of odd characteristics
cites work: Subgroups of Algebraic Groups Containing Regular Unipotent Elements
cites work: Some Representations of Classical Groups
cites work: On the minimal degrees of projective representations of the finite Chevalley groups. II
cites work: Q5645602
cites work: Weil Representations as Globally Irreducible Representations
cites work: Minimal characters of the finite classical groups
cites work: Some generalizations of the Weil representations of the symplectic and unitary groups
cites work: Abstract root subgroups and quadratic action. With an Appendix by A. E. Zalesskii
cites work: Representations of symplectic groups
cites work: Sur certains groupes d'opérateurs unitaires
cites work: The 2-decomposition numbers of Sp(4,q), q odd
cites work: Decomposition numbers of Sp(4,q) for primes dividing \(q\pm 1\)
cites work: Brauer trees of Sp(4,q)
cites work: Q3475470
cites work: Q3810006
cites work: Minimal Polynomials and Eigenvalues of p -Elements in Representations of Quasi-Simple Groups with a Cyclic Sylow p -Subgroup

Cross characteristic representations of symplectic and unitary groups (cites work)
cites work: Permutation representations and a fragment of the decomposition matrix of symplectic and special linear groups over a finite field

Cross characteristic representations of symplectic and unitary groups (Recommended article)
Recommended article: Cross characteristic representations of even characteristic symplectic groups
Similarity Score: 0.9582901
Recommender Run: Recommender Run 3

Cross characteristic representations of symplectic and unitary groups (Recommended article)
Recommended article: Unisingular representations of finite symplectic groups
Similarity Score: 0.9145919
Recommender Run: Recommender Run 3

Cross characteristic representations of symplectic and unitary groups (Recommended article)
Recommended article: Orthogonal, symplectic and unitary representations of finite groups
Similarity Score: 0.9114521
Recommender Run: Recommender Run 3

Cross characteristic representations of symplectic and unitary groups (Recommended article)
Recommended article: Semisimple symplectic characters of finite unitary groups.
Similarity Score: 0.90353644
Recommender Run: Recommender Run 3

Cross characteristic representations of symplectic and unitary groups (Recommended article)
Recommended article: On the irreducibility of symmetrizations of cross-characteristic representations of finite classical groups.
Similarity Score: 0.9003504
Recommender Run: Recommender Run 3

Cross characteristic representations of symplectic and unitary groups (Recommended article)
Recommended article: Cross characteristic projective representations for some classical groups
Similarity Score: 0.8930129
Recommender Run: Recommender Run 3

Cross characteristic representations of symplectic and unitary groups (Recommended article)
Recommended article: Finite Covers, Cohomology and Homogeneous Structures
Similarity Score: 0.89064324
Recommender Run: Recommender Run 3

Cross characteristic representations of symplectic and unitary groups (Recommended article)
Recommended article: Unitary representations, branching rules and matrix elements for the non-compact symplectic groups
Similarity Score: 0.8899572
Recommender Run: Recommender Run 3

Cross characteristic representations of symplectic and unitary groups (Recommended article)
Recommended article: Connections between the representations of the symmetric groups and the symplectic group in characteristic \(2\)
Similarity Score: 0.8877119
Recommender Run: Recommender Run 3

Cross characteristic representations of symplectic and unitary groups (Recommended article)
Recommended article: Unitary Representations of Groups, Duals, and Characters
Similarity Score: 0.88696575
Recommender Run: Recommender Run 3