[GAP Forum] Wedderburn and representation

Angel del Rio adelrio at um.es
Fri Feb 15 22:31:16 GMT 2008

Dear GAP Forum,

This is a reply to a message by Mathias Lederer on Wedderburn
Decomposition of semisimple group algebras and representations.

1) The Wedderburn package enables one to compute decomposition FG = A_1
\times \ldots \times A_s. If I understand correctly, each A_i is not
given in the form A_i = M_{n_i}(D_i) as above. Instead, a cyclotomic
algebra, which is Brauer equivalent to A_i, is given. Can one also
compute the form A_i = M_{n_i}(D_i)? So do we get the division algebra
and the size of the matrices?

To compute the form M_{n_i}(D_i) is usually a difficult task. In fact
just to compute the size of the matrices is not obvious.
Notice that the degree of A_i coincides with the degree of each
irreducible character chi of G which does not vanishes on A_i and it is
equal to n_im_i, where m_i is the Schur index of A_i, or equivalently
the Schur degree of chi over F.
In theory, if F is a global field then the calculation of the Schur
index or even of the Hasse invariant of A_i (or D_i) should be doable by
using local information (See Reiner, Maximal Orders).
Unfortunately in many cases this is a difficult task and as far as I
know there is not an "implementable" method to do this.
This is the subject of many research papers.
See for example:

P. Schmid, Representation-groups for the Schur index, J. Algebra 97
(1985) 101-115.
B. Banieqbal, On bounding the Schur index of induced modules, Bull. LMS
18 (1986) 17-23.
A. Turull, On the Schur index of quasi primitive characters, Journal
LMS  35 (1987)  421-432.
A. Herman, Using character correspondence for Schur index computations,
J. Algebra 159 (2003) 353-360.
A. Herman, Using G-algebras for Schur index computations, J. Algebra 260
(2003) 463-475.

2) Can one compute the multiplicity f_i with which the irreducible
A_i-module shows up in F(G/H)?

Unfortunately there is not anything else that can be said because the
first question is included in the second one for H=1.
Indeed, the multiplicity f_i equals n_i, the size of the matrices and
computing this is already hard for this case.

Resuming, wedderga, provides a description of the simple components of
group algebras, but to have a full understanding of these components we
still need fundamental research on the calculation of Schur indexes and
local invariants.
Some of the experts in the field are Herman, Turull and Schimd.