centralizerSize -- Size of the centralizer of a permutation

Usage:

n = centralizerSize(rho)
Inputs:
- rho, a list,
Outputs:
- n, an integer,

Description

rho is a list representing the cycle type of some permutation: the i-th entry in rho is the number of cycles of length i of the permutation. The output of the function centralizerSize is the size of the centralizer in the symmetric group of any permutation of cycle type rho. If the cycle type rho corresponds to a partition \lambda, then centralizerSize(rho) is also the value of the square norm <p_\lambda, p_\lambda>.

i1 : centralizerSize{1,1,1}

o1 = 6

i2 : R = symmetricRing(QQ,6);

i3 : u = p_1 * p_2 * p_3;

i4 : scalarProduct(u,u)

o4 = 6

o4 : QQ

A few values of z_\rho for small cycle types: the identity of S_n has centralizer size n!, while an n-cycle has centralizer of size n (generated by itself).

i5 : centralizerSize{4}           -- cycle type (1,1,1,1), all of S_4

o5 = 24

i6 : centralizerSize{0,0,0,1}     -- a single 4-cycle

o6 = 4

i7 : centralizerSize{2,1}         -- cycle type (2,1,1)

o7 = 4

Burnside's classical identity \sum_{\lambda \vdash n} 1/z_\lambda = 1 expresses that the uniform measure on S_n sums to 1. We verify this for n = 5:

i8 : parts5 = {{5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1}, {1,1,1,1,1}};

i9 : cycMult = la -> apply(toList(1..max la), i -> #positions(la, j -> j == i));

i10 : sum(parts5, la -> 1/centralizerSize cycMult la)

o10 = 1

o10 : QQ

Ways to use centralizerSize:

centralizerSize(List)

For the programmer

The object centralizerSize is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:7665:0.

centralizerSize -- Size of the centralizer of a permutation

Description

See also

Ways to use centralizerSize:

For the programmer