It is known that the symplectic group Sp2n(p) has two (complex conjugate) irreducible representations of degree (pn + 1)/2 realized over ℚ(√-p), provided that p ≡ 3 mod 4. In the paper we give an explicit construction of an odd unimodular Sp2n(p) · 2-invariant lattice Δ(p, n) in dimension pn + 1 for any pn ≡ 3 mod 4. Such a lattice has been constructed by R. Bacher and B. B. Venkov in the case pn = 27. A second main result says that these lattices are essentially unique. We show that for n ≥ 3 the minimum of Δ(p, n) is at least (p + 1)/2 and at most p(n - 1)/2. The interrelation between these lattices, symplectic spreads of double-struck F sign2np, and self-dual codes over double-struck F signp is also investigated. In particular, using new results of U. Dempwolff and L. Bader, W. M. Kantor, and G. Lunardon, we come to three extremal self-dual ternary codes of length 28.
ASJC Scopus subject areas
- Algebra and Number Theory