The group action on the range of functions |

is compatible with the group action sincej:Y^{n}-> Y^{n-1}, f -> j(f):=f |_{n-1}

From that we deduce that a transversalj(h o f)=(h o f) |_{n-1}=h o f |_{n-1}= h o j(f).

whereT(H\\Y^{n})=È_{f'ÎT(H\\Yn-1)}T(H_{f'}\\j^{-1}({f'} ),

acts on the setH_{f'}={hÎH|h o f'=h)} ={hÎH|hf'(i)=f'(i) " iÎn-1)}

j^{-1}({f'} )={fÎY^{n}|j(f)=f')} ={fÎY^{n}|f |_{n-1}=f')} .

This method leads to the following recursive algorithm:
Without loss of generality let *Y* be * m*,
then each function

- For
*f(1)*take any element of*T(H\\Y)*. - Let
*i<n*and let the first*i*values of the canonic representative be*(f(1),...,f(i))*, then determine the pointwise stabilizer*H*of the set_{{f(1),...,f(i)} }*{f(1),...,f(i)}*.

and take for*H*_{{f(1),...,f(i)} }={hÎH_{{f(1),...,f(i-1)} }|hf(i)=f(i))}*f(i+1)*any element of*T(H*._{{f(1),...,f(i)} }\\Y)

In this situation we can also compute the canonic representative of
the orbit *H(f)*
from any orbit element *f=(f(1),...,f(n))Î m^{n}*.
Let

thenH_{i}={hÎH_{i-1}|hf_{i}(i)=f_{i}(i))} ,

This algorithm can easily be changed to produce only representatives of
injective functions. In the case *n=m* these representatives are coset
representatives *H\S _{n}*.

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001

The group action on the range of functions |