Matrices may be regarded as defining a design with the entry of a matrix in a particular row and column defining the incidence of the row and column. The automorphism group of a matrix is the set of permutations of the rows and columns of the matrix that leave the matrix unchanged.
The action of a permutation on a matrix by permuting rows and columns. If M has r rows and c columns then x must have degree r + c and fix the set R = {1..r}. The action of x on R gives the permutation of the rows of the matrix. The remainder of x gives the action on columns.
Computes the group of all permutations x such that Mx= M. Currently this is done by constructing a graph from M, and applying the AutomorphismGroup function to the graph.
Finds a permutation x such that Mx= N, if such exists. If a permutation is found return values are true and the permutation, otherwise returns false. Currently this is done by constructing graphs from M and N, and applying the IsIsomorphic function to the graphs.
> M := VertexFacetHeightMatrix(PolytopeSmoothFanoDim3(10)); M; [1 0 0 0 2 1 2 2] [0 0 0 1 1 2 2 2] [0 0 1 2 0 2 2 1] [2 0 1 0 2 0 2 1] [0 2 0 1 1 2 0 2] [1 2 0 0 2 1 0 2] [0 2 1 2 0 2 0 1] [2 2 1 0 2 0 0 1] [2 2 2 1 1 0 0 0] [1 2 2 2 0 1 0 0] [1 0 2 2 0 1 2 0] [2 0 2 1 1 0 2 0] > A := AutomorphismGroup(M); A; Permutation group A acting on a set of cardinality 20 Order = 24 = 2^3 * 3 (2, 4)(3, 12)(5, 8)(7, 9)(13, 18)(15, 16)(17, 20) (1, 5)(2, 6)(3, 8)(4, 7)(9, 11)(10, 12)(13, 16)(14, 19)(17, 18) (1, 2)(3, 4)(5, 6)(7, 8)(9, 10)(11, 12)(13, 16)(17, 18) > Nrows(M), Ncols(M); 12, 8 > Orbits(A); [ GSet{@ 14, 19 @}, GSet{@ 13, 18, 16, 17, 15, 20 @}, GSet{@ 1, 5, 2, 8, 6, 4, 3, 7, 12, 9, 10, 11 @} ]The automorphism group is transitive on the rows of the matrix. Now dualize the matrix and test for isomorphism.
> D := Matrix(12, 8, [2-x:x in Eltseq(M)]); > f, x := IsIsomorphic(M, D); f, x; true (2, 4)(3, 12)(5, 8)(7, 9)(14, 19)(15, 17)(16, 20) > M^x eq D; true[Next][Prev] [Right] [Left] [Up] [Index] [Root]