Abstract:
It is known that the symplectic group $Sp_{2n}(p)$ has two (complex
conjugate) irreducible representations of degree $(p^{n}+1)/2$ realized
over $\QQ(\sqrt{-p})$, provided that $p \equiv 3 \bmod 4$. In this paper
we give an explicit construction of an odd unimodular $Sp_{2n}(p) \cdot
2$-invariant lattice $\Delta (p,n)$ in dimension $p^{n}+1$ for any $p^{n}
\equiv 3 \bmod 4$. Such a lattice has been constructed by R. Bacher --
B. B. Venkov in the case $p^{n} = 27$. A second main result says that these
lattices are essentially unique. We show that for $n \geq 3$ the minimum
of $\Delta(p,n)$ is at least $(p+1)/2$ and at most $p^{(n-1)/2}$. The interrelation
between these lattices, symplectic spreads of $\FF^{2n}_{p}$, and self-dual
codes over $\FF_{p}$ is also investigated. In particular, using new results
of U. Dempwolff and L. Bader -- W. M. Kantor -- G. Lunardon, we come to
three extremal self-dual ternary codes of length $28$.
Key words and phrases: Integral lattice, unimodular lattice, integral representations, finite symplectic group, Weil representation, symplectic spread, linear code, self-dual code
1991 Mathematics Subject Classification: Primary 20C10; Secondary 11E12, 20C20, 51E23, 94B05.
Published in: Journal of Algebra 194, 113-156 (1997)
Contact: Rudolf.Scharlau@Mathematik.Uni-Dortmund.DE