The package contains Magma implementations for some recognition algorithms for small-dimensional representations of classical groups.
The two main functions are RecogniseAltSquare and RecogniseSymSquare.
intrinsic RecogniseAltSquare( G::GrpMat : type := "SL", CheckResult := true, Method := "Recursive" ) -> BoolElt, Map, Map, GrpMatElt, GrpMatElt
G should be matrix group conjugate to the exterior square representation of SL( d, q ). Returns true/false, a map from SL( d, q ) to G, a map from G to SL( d, q ), and a matrix whose rows form a basis that exhibits the alt square structure. The last piece of output can be used to conjugate the image of G under the second map into the standard copy of the classical group in Magma.
Supply CheckResult := true to check the final result.
he basic algorithm is implemented in two variations. The first uses a recursive call for smaller dimensional exterior square recognition, while the second uses recognition of tensor decomposition with IsTensor. In small dimensions (how small depends on the type of the group), the version using tensor recognition is called, while if the dimension is high enough, then the recursive version is used. This choice can be overwritten by setting to "Tensor".
intrinsic RecogniseSymSquare( G::GrpMat : type := "SL", CheckResult := false ) -> BoolElt, Map, Map, GrpMatElt, GrpMatElt
G should be matrix group conjugate to the symmetric square representation of SL( d, q ). Returns true/false, a map from SL( d, q ) to G, a map from G to SL( d, q ), and a matrix whose rows form a basis that exhibits the sym square structure. The last piece of output can be used to conjugate the image of G under the second map into the standard copy of the classical group in Magma.
Supply CheckResult := true to check the final result.
The file smalldimreps.m contains an implementation of Greenhill's algorithm to recognize if a matrix is the alternating square of another matrix.
The algorithm is described in C Greenhill, "An algorithm for recognising the exterior square of a matrix", Linear and Multilinear Algebra, 1999.
The function assumes that the input is a non-singular matrix that is a member of AltSquare( SL( d, q )) in the basis e_12, e_13,...,e_1d,...,e_{d-1}d.
Greenhill's paper describes the algorithm for arbitrary, not necessary invertible, matrices, but this implementation only works for invertible matrices and the other cases may lead to error!
Written by Csaba Schneider. [email protected] www.mat.ufmg.br/~csaba