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Next: Lie Theory
Up: Algebras
Previous: Characters of Finite Groups
New Features:
- A key problem when constructing an ordinary irreducible representation
tion of a group is to determine its Schur index, that is, the degree of
a minimal field for the representation taken over the field generated
by its character values. The first practical algorithm for this was
developed in 2006 by G. Nebe and W. Unger. This version of Magma contains
an implementation of the algorithm. The algorithm works with characters of
the group and its subgroups, and fields generated by character values
to determine the Schur index of the character over
all completions of the rationals. C. Fieker has provided a routine that
uses this data to compute the Schur index of the character over a given
number field.
The algorithm first computes strong upper bounds for the
local Schur indices, then uses the Brauer-Witt Theorem and a search through
subgroups of the group to reduce to considering quasi-elementary groups.
Schur indices are then computed by calculations with values of Brauer
characters, degrees of field extensions and, in the 2-adic case, further
reduction to a case considered by
U. Riese & P. Schmid. The upper bounds used are sufficiently strong that
in many cases the subgroup search, which may be very time-consuming,
is not necessary.
The names of the new intrinsic functions are SchurIndex, for the
Schur index over a particular number field, and SchurIndices for
the indices over all completions of a number field.
- The problem of writing a given (absolutely irreducible) representation
over as small a field as possible (or over an ``arbitrary'' user defined
field) is a key problem in representation theory. A new method due to
C. Fieker and based on Galois cohomology has been implemented. This
method will find a minimal subfield that affords a given representation.
If this field is not ``small enough'' then a constructive version of the
Grunwald-Wang theorem is used to find a minimal degree splitting field.
Next: Lie Theory
Up: Algebras
Previous: Characters of Finite Groups
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