# Peter Webb: Undergraduate Research Projects

Spring 2006: Robert Hank and Bryan Simpkins worked on a project to create software for computing with representations of finite groups over fields of positive characteristic, written within the language GAP. They started by updating some old routines which Webb had written in a previous version of GAP, and this gave some experience with the system. They then worked on implementing and refining an algorithm to find a complete list of indecomposable summands of a representation. This routine is now part of the collection of routines available from
http://www.math.umn.edu/~webb/GAPfiles/
They also wrote a routine to compute the matrix which represents the sum of all the group elements. This, too, is part of the collection of routines.

January 2007: Brad Froehle wrote a routine SymmetricPowerRep which returns the representation on a specified symmetric power of a given group representation.

March 2011: Fan Zhang wrote a routine Spin as part of the package 'catreps' which computes with representations of categories in the same way that the package 'reps' computes with group representations. These packages are written in the language GAP. The routine Spin computes a basis for the representation space generated by a given set of vectors which lie in a given representation of a category. Representations are notationally more complex than those of groups, and much of the work in implementing algorithms for categories is in dealing with this complexity.

July 2018: Moriah Elkin wrote several routines as part of the package 'catreps'. She upgraded the routine ConcreteCategory so that it now accepts either one or two arguments, the second being a list of the objects in the category. Previously the objects were deduced as the domains of the morphisms in the first argument. She added some checks to the code. New routines: EndomorphismGroups(cat) creates a field listing the endomorphism groups of objects under the assumption they are groups, and that their generators appear among the generators of the category. FI(n) returns the category of finite sets of size at most n with injective maps. DirectSumRep(cat, rep1, rep2) returns the representation that is the direct sum of the representations rep1 and rep2 of the category cat. SubConstant(rep) returns a list of bases for the largest subconstant representation of rep. It calls routines MorphismsRep and GeneratorDomains.