Symplectic groups, symplectic spreads, codes, and unimodular lattices
type: Item
instance of: scholarly article
title: Symplectic groups, symplectic spreads, codes, and unimodular lattices
zbMATH Open document ID: 0887.20004
author: Rudolf Scharlau
author: Pham Huu Tiep
published in: Journal of Algebra
publication date: 1998-05-26T00:00:00Z
review text: The symplectic group \(\text{Sp}(2n,p)\) has a complex representation of degree \(p^n\) which splits into two irreducible representations, the Weil representations, of degrees \((p^n-1)/2\) and \((p^n+1)/2\) respectively. One of these representations, say \(W\), has even degree and is faithful. Suppose \(p\equiv 3\pmod 4\) then \(G=\text{Sp}(2n,p)\cdot 2\) has a faithful \(\mathbb{Q}\)-module \(V\) of degree \(p^n+1\) which splits as an \(\text{Sp}(2n,p)\)-module in \(W\) and \(\overline W\) (complex conjugate representation). As the first main result the authors construct in \(V\) a \(G\)-invariant unimodular lattice. In the second main result it is shown that for \(p^n\equiv 3\pmod 4\) this unimodular lattice is unique as a faithful, irreducible, unimodular \(G\)-lattice of degree \(p^n+1\) and that any faithful, irreducible \(G\)-lattice of degree \(p^n+1\) is similar to this lattice or one of two other closely related lattices. The authors handle the case \(p^n\equiv 1\pmod 4\) in a successor of this paper. Finally the following application is given: The construction of the unimodular lattice exhibits an interesting correspondence between selfdual codes of length \(p^3+1\) over \(\text{GF}(p)\) and symplectic spreads in \(\text{GF}(p)^6\). It is well known that for each odd prime \(p\) there are at least two nonisomorphic symplectic spreads in \(\text{GF}(p)^6\) which proves the existence of two selfdual codes of length \(p^3+1\) over \(\text{GF}(p)\). Related work on lattices and Weil representations of \(\text{Sp}(2n,p)\) is [\textit{R. Gow}, J. Algebra 122, No. 2, 510-519 (1989; Zbl 0679.20039)]\ and [\textit{B. H. Gross}, J. Am. Math. Soc. 3, No. 4, 929-960 (1990; Zbl 0745.11035)].
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 11H56
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 51E25
zbMATH DE Number: 1053878
zbMATH Keywords: symplectic groups
zbMATH Keywords: irreducible representations
zbMATH Keywords: Weil representations
zbMATH Keywords: unimodular lattices
zbMATH Keywords: selfdual codes
zbMATH Keywords: symplectic spreads
reviewed by: Ulrich Dempwolff
MaRDI profile type: Publication
OpenAlex ID: W1976106040
cites work: Lattices and association schemes: a unimodular example in dimension 28
cites work: Some extreme forms defined in terms of Abelian groups
cites work: Q4323109
cites work: On the Clifford collineation, transform and similarity groups. I.
cites work: On lattices equivalent to their duals
cites work: Q5688167
cites work: Translation planes of order 27
cites work: Q3997911
cites work: Symplectic group lattices as Mordell-Weil sublattices
cites work: Q4717323
cites work: On Integral Representations of Finite Groups
cites work: The Growth of Class Numbers of Quadratic Forms
cites work: Even unimodular lattices associated with the Weil representation of the finite symplectic group
cites work: Group Representations and Lattices
cites work: Eine nicht-desarguesche zweifach transitive affine Ebene der Ordnung 27
cites work: On extremal self-dual ternary codes of lengths 28 to 40
cites work: Characters of Solvable and Symplectic Groups
cites work: Klassenzahlen definiter quadratischer Formen
cites work: Orthogonal decompositions and integral lattices
cites work: An upper bound for self-dual codes
cites work: Q4146667
cites work: On norm three vectors in integral Euclidean lattices. I
cites work: Modular lattices in Euclidean spaces
cites work: Q4203460
cites work: Symplectic group lattices
cites work: Some Representations of Classical Groups
cites work: Globally irreducible representations of finite groups and integral lattices
cites work: Globally irreducible representations of the finite symplectic group Sp4(q)
cites work: Q3722747
cites work: Representations of symplectic groups
cites work: Quadratic residue codes in their prime
cites work: Transformations of a Quadratic Form Which Do Not Increase the Class-Number
cites work: Q3972837
DOI: 10.1006/JABR.1996.6902
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description: scientific article; zbMATH DE number 1053878