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Automorphisms of Matrices

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.

M ^ x : Mtrx, GrpPermElt -> Mtrx
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.
AutomorphismGroup(M) : Mtrx -> GrpPerm
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.
IsIsomorphic(M, N) : Mtrx, Mtrx -> BoolElt, GrpPermElt
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.

Example Design_FanoAuto (H147E13)

We construct a matrix from a Fano polytope and compute its automorphism group.

> 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
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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013