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GAP Forum

On Fri, Aug 03, 2001 at 09:48:34AM -0400, Vivek Shende wrote:

> Does the function PermutationMat have an inverse? that is, if i have a matrix

> that is the matrix of a permutation, is there some function that will give me

> that permutation?

Dear Vivek,

In fact, `Permutation' does what you want, though the documentation

does not currently make this clear e.g.

gap> g := PermutationMat((1,2,3,4)(5,6,7), 10); [ [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ]

returns a 10x10 matrix that permutes the 10 standard basis (row) vectors

of Q^n (Q = Rationals), which, since GAP doesn't really distinguish a

matrix from a list of row vectors, is given by:

gap> xset := PermutationMat((), 10); #among other ways [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ] If we identify each element of xset with its position index i.e. [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] is 1, [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] is 2, etc. then `Permutation( <g>, <xset> );' returns the permutation that corresponds to the action of <g> on <xset> in terms of <xset>'s position indices i.e. since <g> takes the 1st <xset> vector to the 2nd <xset> vector etc. `Permutation( <g>, <xset> );' will start `(1, 2, ...'. So: gap> Permutation(g, xset); (1,2,3,4)(5,6,7)

I'll see what I can do to improve the documentation here for the next

release.

[The example given in the current documentation is:

gap> Permutation((1,2,3)(4,5)(6,7),[4..7]); (1,2)(3,4) One might have thought the result should be (4,5)(6,7) as the action of (1,2,3)(4,5)(6,7) restricted to the points [4,5,6,7] is (4,5)(6,7), but `Permutation' returns a permutation in terms of the position indices of the points 4 .. 7 in the list [4,5,6,7].] Regards, Greg Gamble ___________________________________________________________________ Greg Gamble __________________ mailto:gregg@math.rwth-aachen.de Lehrstuhl D fuer Mathematik Tel: +49 241 804545 Templergraben 64 52062 Aachen, Germany http://www.math.rwth-aachen.de/~Greg.Gamble ___________________________________________________________________

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