Reflective integral lattices
by Rudolf Scharlau and Britta Blaschke
Abstract:
A lattice $L$ with a positive definite quadratic form is called reflective if
the unique largest subgroup generated by reflections
of the orthogonal group ${\mathrm O}(L)$ has no fixed vector. Equivalently,
the ``root system'' $\sR(L)$ has maximal rank. The root system of a
lattice is defined
in Section 1 below; the roots are not necessarily of length 1 or 2.
In Section 2 the structure of reflective lattices is worked out. They
are described and classified by pairs $(\sR,\CL)$, where $\sR$ is
a ``scaled root system'', and the ``code'' $\CL$ is a subgroup of
the ``reduced discriminant group'' $\overline{T}(\sR)$.
The crucial point is that $\overline{T}(\sR)$ only depends on the
combinatorial equivalence class of the root system $\sR$.
In Section 3 we give
a precise description of the full root system of a reflective lattice if
one starts with a sub root system of combinatorial type $n\sA_1$ or
$m\sA_2$. In Section
4 our techniques are applied to a complete and explicit description of all
reflective lattices in dimensions $\le 6$.
Key words and phrases: lattice, integral quadratic form,
root system, reflective lattice, binary code, glue code,
Weyl group.
1991 Mathematics Subject Classification: Primary 11H55; Secondary 11E12, 51F15, 94B05.
Published: J. of Algebra 181, 934-961 (1996)
Contact: Rudolf.Scharlau@Mathematik.Uni-Dortmund.DE or Blaschke@Mathematik.Uni-Bielefeld.DE