## 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 _{G}X 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 Î*__m__,j Î__n__.

This can be seen as follows:
Remember the direct factors * S*_{ m} ^{j}, for *j Î*__n__, of the base
group * S*_{ m} ^{*} of * S*_{ m} wr S_{ n} (cf. the remark on *H*^{x}
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.

last changed: January 19, 2005