Take the 2-minute tour ×  Question 1 Given a representation of a
finite group, what algorithm can be used to check is it
irreducible or not ?  (Main case - complex numbers, comments on
other cases are also welcome. "Given" means finite set of
matrices is given).  Question 2 Given a representation of a
finite group, what algorithms can be used to decompose it to the
direct sum of irreducibles) ?  For the question 1 I would do the
following: rep is irrep if its commutant consists of scalar
matrices. So I can try to find matrices commuting with all
elements of the group and look whether I get only scalar
matrices.  Are there more effective ways to do it ?  Related
question: How to compute all irreducible representations of a
finite group ? (how GAP is doing this?)  share|improve this
question 1) Compute the norm of its character. 2) Compute the
character table of the group, then the character of the
representation you're looking at. This is basic material in the
representation theory of finite groups and not app