Weil representations of the symplectic group
type: Item
instance of: scholarly article
title: Weil representations of the symplectic group
zbMATH Open document ID: 0932.20047
author: Fernando Szechtman
published in: Journal of Algebra
publication date: 2000-01-12T00:00:00Z
review text: Let \(q\) be an odd prime power and \(\text{Sp}:=\text{Sp}_{2n}(q)\) be the finite symplectic group over the field of \(q\) elements. A Weil representation of \(\text{Sp}\) is a complex representation \(R\) of degree \(q^n\), which decomposes as the sum of two irreducible constituents, \(R_0\) of degree \((q^n+1)/2\) and \(R_1\) of degree \((q^n-1)/2\). Weil representations have been the subject of intensive research since the 1960's. The purpose of the paper is to give explicit constructions of Weil representations. \(\text{Sp}\) acts on the Heisenberg group \(H\) of order \(q^{2n+1}\) and acts trivially on the centre of \(H\). Hence any irreducible complex representation of \(H\) of degree \(q^n\) is stabilized by \(\text{Sp}\) and thus gives rise to a projective representation \(R'\) of \(\text{Sp}\). Having constructed an explicit model of \(R'\), the author shows that \(R'\) is equivalent to an ordinary representation \(R\), without referring to known results about the Schur multiplier of \(\text{Sp}\). Next he shows that \(R_0\) and \(R_1\) can be realized over the smallest possible field, namely its character value field \(\mathbb{K}\), except for the case of \(R_1\) with \(\mathbb{K}\subseteq\mathbb{R}\), where \(R_1\) can be realized over a quadratic extension of \(\mathbb{K}\). Again, his construction is explicit and does not require Schur index calculations.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20C25
zbMATH DE Number: 1230357
zbMATH Keywords: finite symplectic groups
zbMATH Keywords: Weil representations
zbMATH Keywords: explicit constructions
zbMATH Keywords: irreducible complex representations
zbMATH Keywords: projective representations
reviewed by: Pham Huu Tiep
MaRDI profile type: Publication
OpenAlex ID: W1977270628
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DOI: 10.1006/JABR.1998.7516
description: scientific article; zbMATH DE number 1230357