Examples |

A numerical example is provided by *S _{ 3}
wr S_{ 2}*. The set of proper
partitions characterizing the conjugacy classes of

{ a | a|¾2 }= {(2),(1^{2}) },

the set of corresponding cycle types is

{a | a |¾| 2 }= {(0,1),(2,0) }.

Thus the types of *S _{ 3} wr S_{ 2}*
turn out to be

0 | 1 |

0 | 0 |

0 | 0 |

0 | 0 |

0 | 1 |

0 | 0 |

0 | 0 |

0 | 0 |

0 | 1 |

2 | 0 |

0 | 0 |

0 | 0 |

0 | 0 |

2 | 0 |

0 | 0 |

0 | 0 |

0 | 0 |

2 | 0 |

1 | 0 |

1 | 0 |

0 | 0 |

1 | 0 |

0 | 0 |

1 | 0 |

0 | 0 |

1 | 0 |

1 | 0 . |

The orders of the conjugacy classes are 6,18,12,1,9,4,6,4,12.
We now describe an interesting action of * S _{ m} wr S_{ n} * which
is in fact an action of the form

Example:The action ofSon_{ m}wr S_{ n}is obviously similar to the following action ofmnSon the set_{ m}wr S_{ n}:m´nThe corresponding permutation group onS_{ m}wr S_{ n}´(m´n) ->m´n:(( y, p),(i,j)) -> ( y( pj)i, pj).will be denoted bym´nand called theS_{ n}[ S_{ m}]compositionofSand_{ n}S, while_{ m}will be used for the permutation group onG[H]Y ´X, induced by the natural action ofH wron_{X}GY ´X. The action of the wreath productSon_{ m}wr S_{ n}induces a natural action ofm´nSon the set_{ m}wr S_{ n}i.e. on the set of 0-1-matrices consisting ofY^{X}:=2^{ m´n}= {(a_{ij}) | a_{ij}Î{0,1 },i Îm,j În},mrows andncolumns:SinceS_{ m}wr S_{ n}´2^{ m´n}:(( y, p),(a_{ij})) -> (a_{ y-1(j)i, p-1 j}).( y, p)=( y,1)( i, p), we can do this in two steps:Hence we can first of all permute the columns of(a_{ij}) -> (a_{i, p-1j}) -> (a_{ y-1(j)i, p-1j}).(ain such a way that the numbers of 1's in the columns of the resulting matrix is nonincreasing from left to right:_{ij})åAnd after having carried out this permutation with a suitable_{i}a_{i, p-11}³å_{i}a_{i, p-12}³...p, we can find ayÎSthat moves the 1's of each column in flush top position. This proves that the orbit of_{ m}^{*}(aunder_{ij})Sis characterized by an element of the form_{ m}wr S_{ n}(which is an element of

1 ... ... ... 1 . . 1 ... 1 0 2), i.e. by a proper partition of^{ m´n}k:= å. Hence the orbits of_{i,j}a_{ij}Son_{ m}wr S_{ n}2are characterized by the proper partitions^{ m´n}a, where each partaand where the total number of parts is_{i}£n£m:Hence an application of the Cauchy-Frobenius Lemma yields the following formula for the number of partitions of this form:Corollary:There exists a natural bijectionS_{ m}wr S_{ n}\\2^{ m´n}-> { a|¾k | k <= mn, a_{1}<= n, l( a) £m }.which can be made more explicit by an application of the Lemma.| S_{ m}wr S_{ n}\\2^{ m´n}| = (m!^{n}n!)^{-1}å_{( y, p) ÎS m wr S n }2^{ S n c(h n( y, p))},

harald.fripertinger "at" uni-graz.at | http://www-ang.kfunigraz.ac.at/~fripert/ |

UNI-Graz | Institut für Mathematik |

UNI-Bayreuth | Lehrstuhl II für Mathematik |

Examples |