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GAP 3 Share Package "arep"AREP: Abstract REPresentationsShare package since August 1998, communicated by Herbert Pahlings. This package has not yet been transferred to or replaced in GAP 4. You can use it only in GAP 3. AuthorsSebastian Egner and Markus Püschel. Implementation
Language: GAP 3 (includes a C program which speeds
up some computations) DescriptionThe package AREP was developed to create data structures and functions for the efficient calculation with group representations with special attention to permutation and monomial representations. The idea is to calculate with representations up to equality not only up to equivalence as it is usually done in representation theory. Furthermore we wanted to be able to create highly structured representations in GAP 3 as you do it on a piece of paper by writing e.g. R = (R1 innerTensorProduct R2) induction G,
where R1, R2 are representations and G is a group. The representation
R constructed this way should be kept this way; it should not be
evaluated until you ask for it.
In this sense we have implemented an infrastructure for representations.
The more advanced part is given by functions which allow to transform
a given representation. These functions are mostly decompositions in
a certain sense. E.g. it is well known that every transitive monomial
representation of a group G is equivalent to an induction of a
onedimensional representation of a subgroup H <= G. But how do
you choose the subgroup, the transversal and the conjugation to obtain
equality? This problem is solved by the function
R = (L induction_By_T G)^D,
where L is a onedimensional rep of a subgroup. A more sophisticated
function for constructively decomposing a given representation
is the function
R = (R1 directSum R2 ... Rk)^A
where A striking application of constructive representation theory is the decomposition of matrices with symmetry. This technique allows to explain and generate a lot of classical fast signal transforms. The original idea is due to Minkwitz and was further developed by the authors. Abstractly, a symmetry of a given matrix M is given by a pair (R1, R2) of representations of the same group G, such that R1(g) * M = M * R2(g) for all g in G.
If R1 and R2 both are permutation representations we call
the symmetry PermPermSymmetry. MonMonSymmetry is defined
analogous. Functions to determine the symmetry of a given matrix
can be found in cd lib/ grep #F complex.g tells you about the functions in 'complex.g'. We recommend to extract these documentations before you proceed with the examples below. ManualAn AREP manual is available as dvi, or pdf, or postscript file. Contact addresses
Sebastian Egner 