Complete monomial groups

Complete monomial groups

We have already met the wreath product H wr X G , where G is a group acting on X while H acts on Y. Now we consider the particular case where G is a permutation group, say G <= S n , and where we take for GX the natural action of G on n. In this case we shorten the notation by putting

H wr G:=H wr n G.

A particular case is H wr S n , the complete monomial group of degree n over H. Many important groups are of this form, examples will be given in a moment. In the case when H <= S m , then H wr G has the following natural embedding into S mn :

d: S m wr S n hookrightarrow S mn

where (y,p) is mapped onto the permutation given by

(j-1)m+i -> ( pj-1)m+ y( pj)i        " i Îm,j În.

This can be seen as follows: Remember the direct factors S m j, for j În, of the base group S m * of S m wr S n (cf. the remark on Hx in the Lemma). Its image d[ S m j] acts on the block {(j-1)m+1, ..., jm } as S m does on m, while the image d[ S n '] of the complement S n ' of the base group acts on the set of these n subsections {(j-1)m+1, ...,jm } of length m of the set mn as S n does act on n. For example the element

( y, p):=( y(1), y(2), y(3), p):=((12),(123),1, (23)) ÎS 3 wr S 3

is mapped under d onto

(12)(456) (47)(58)(69) =(12)(475869) ÎS 9,

where (12)(456)= d(( y,1)) and (47)(58)(69)= d(( i, p)). The image of H wr G under d will be denoted as follows:

H G:= d[H wr G].

It is called the plethysm of G and H, for reasons which will become clear later.

 harald.fripertinger "at" uni-graz.at http://www-ang.kfunigraz.ac.at/~fripert/ UNI-Graz Institut für Mathematik UNI-Bayreuth Lehrstuhl II für Mathematik
last changed: January 19, 2005

 Complete monomial groups