[GAP Forum] Wedderburn and representations
Mathias Lederer
mlederer at math.uni-bielefeld.de
Thu Feb 14 22:23:44 GMT 2008
Dear GAP forum,
I am currently learning to use the Wedderburn package, which I find
very appealing. I want to apply the Wedderburn also to the following
situation, which deals with a special class of representations of G.
Take a group G and a field F such that FG is semisimple. Let FG = A_1
\times \ldots \times A_s be the Weddderburn decomposition of FG.
Next, let H be a subgroup of G. The vector space
F(G/H) := \oplus_{g \in G} F gH
(that is, a vector space with the coset classes as a basis) has a
canonical action of G, hence, is an FG-module. Therefore, it is a
direct sum V_1 \oplus \ldots \oplus V_s, where V_i is a module over
A_i. The algebra A_i is a matrix algebra over a division algebra,
say, A_i = M_{n_i}(D_i). Up to isomorphism, there exists a unique
irreducible A_i-module, to wit, U_i = D_i^{n_i}. Hence the module V_i
is a direct sum of a number of copies of U_i, say, V_i = U_i^{f_i}.
Here are my questions.
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?
2) Can one compute the multiplicity f_i with which the irreducible
A_i-module shows up in F(G/H)?
Many thanks in advance,
Mathias
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