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Next: Character Theory
Up: Representation Theory
Previous: Modules over Algebras
New Features:
- An algorithm has been developed for computing irreducible
Q[G]-modules for a finite group G. Given a rational character
of G, the algorithm proceeds by locating a (reducible) module that
contains the desired module. Then using the Meataxe described above,
the module M is split thereby yielding the required irreducible
module. Use is made of condensation to reduce the dimensions
of the modules that have to be split. The algorithm controls the
growth of coefficients at every stage, thus returning modules whose
actions are usually defined by matrices with very small integral
entries. A variant of the algorithm is provided which determines
all irreducible
Q[G]-modules for G. The machinery has been used
to construct irreducible
Q[G]-modules having dimension well over
a thousand in favourable circumstances. New functions:
- IrreducibleModules(G, RationalField())
to compute all or some irreducible modules for G over
Q
(with many options).
- RationalCharacterTable(G)
to compute the table of irreducible rational characters for G.
- GModule(chi, RationalField()): compute irreducible module
for given irreducible rational character.
- A specialised method for splitting a large-degree permutation module to obtain
a specific irreducible has been included. The algorithm is a combination of the
Michler-Weller algorithm for determining character values of constituents of a
permutation representation, together with Nickerson's ``Split-P" condensation
method. The character values are used to identify the correct module to be
uncondensed to obtain the G-module affording the given character. The results
of Michler and Weller allow an algorithmic search for the right vector to spin,
as opposed to Nickerson's heuristic approach.
Next: Character Theory
Up: Representation Theory
Previous: Modules over Algebras
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