Radicals of modules using linear algebra Let $\mathcal{A}$ be an associative algebra over a field $F$ with generators $a_1,a_2,\ldots,a_n$ and let $M$ be an $\mathcal{A}$-module. We can specify $M$ with the following data:


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*a $d$-dimensional $F$-vector space 

*$n$ $d\times d$ matrices which describe the actions of $a_1,\ldots,a_n$ on $M$. 


How do we compute (theoretically) the radical of $M$ (that is, the intersection of all its maximal $\mathcal{A}$-submodules) using this data and linear algebra? 
 A: GAP finds the socle of the dual module. The method is a little silly: find polynomials to recognize all the simple factors in the module, then recognize simple submodules using them. I'm not sure if this is group algebra specific (I kind of think it works in general for finite dimensional algebras), but you might be careful in assuming such recognition polynomials exist in general. For instance, if you want the isotypic compontent of trivial (central) modules, then a family of polynomials (which is good enough) is $a_i−a_i^0$ for generators $a_i$ of the algebra. Their simultaneous kernel is a summand of the socle.
Generally speaking, this is called the "meataxe". See page 230ff sections 7.4.1 and 7.5.2 of Holt's Handbook of CGT. Typically the family has only one member, but may include more than exactly the isotypic component of the socle, and so an additional step is required, but the additional step is quick. I can't find the description of how "chop" chooses its one true polynomial, but if I recall correctly, it has the property that no other isoclass of module lies in its nullspace.


*

*Holt, Derek F.; Eick, Bettina; O'Brien, Eamonn A.
Handbook of computational group theory.
Chapman & Hall/CRC, Boca Raton, FL, 2005. xvi+514 pp. ISBN: 1-58488-372-3
MR2129747
DOI:10.1201/9781420035216
