Random distribution of words

The group algebra can also be used to investigate the random distribution of words of a certain length in the generators of the group.

> M11 := sub< Sym(11) | (1,11,9,10,4,3,7,2,6,5,8), (1,5,6,3,4,2,7,11,9,10,8) >;
> A := GroupAlgebra(RealField(16), M11 : Rep := "Vector");
> A;
Group algebra with vector representation
Coefficient ring: Real Field of precision 16
Group: Permutation group M11 acting on a set of cardinality 11
    Order = 7920 = 2^4 * 3^2 * 5 * 11
        (1, 11, 9, 10, 4, 3, 7, 2, 6, 5, 8)
        (1, 5, 6, 3, 4, 2, 7, 11, 9, 10, 8)
> e := (A!M11.1 + A!M11.2) / 2.0;
> eta := Eta(A) / #M11;

For growing n, the words of length n in the generators of M11 converge towards a random distribution iff converges towards eta. We look at the quadratic differences of the coefficients of eta for n = 10,20,30,40,50.

> e10 := e^10;
> f := A!1;
> for i in [1..5] do
>     f *:= e10;
>     print &+[ c^2 : c in Eltseq(f - eta) ];
> end for;
0.0012050667195213
1.289719354694155e-5
5.9390965208879e-7
3.394099291966e-8
2.19432454574986e-9

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