Subject: Ravenel's 2nd invariant theory question
From: Peter Webb <webb@math.umn.edu>
Date: Wed, 3 Aug 2005 00:20:32 -0500 (CDT)

Dear Topology List,

I, too, am very intrigued by the phenomenon which Doug has presented to 
us. 
I had been emailing Doug separately about this, but it seems that a number 
of people are interested and it is clearly the thing to send comments 
round 
to everyone.

The thing which amazes me about Doug's observation is that he shows that 
the 
reciprocal of the Poincare series of the symmetric powers of the symmetric 
group S_3 acting on {1,2,3} is a polynomial with coefficients in the 
Burnside 
ring. I have never seen such an expression before, and I think it would be 
extremely interesting to know if this occurs more generally.

As has been pointed out, I produced a formula for this Poincare series in 
my paper in the Oaxtepec proceedings, and this is quite good (in my view!) 
for calculating the series in particular cases (including the S_3 
example), 
but is troublesome as a means to attack the present problem at a more 
theoretical level. One trouble is that the only G-sets which occur in the 
Poincare series for a symmetric group acting naturally (for example) are 
Young subgroups, and this is not obvious from my formula.

In this connection I sent the following comments to Doug this morning 
about 
the generality in which one might work. We may consider any finite group G 
permuting a set O. By a composition of O I will mean a collection of 
disjoint subsets of O whose union is O. Thus O = O_1 \cup ... \cup O_n 
is a disjoint union, and I want to ignore the order of the O_i . For 
each such composition C let G_C be the set of g in G which preserve 
each O_i. Thus in case G is a symmetric group on n symbols and O is 
{1 ... n} these subgroups are the Young subgroups.

Proposition. Let O be a G-set. The collection of subgroups G_C as C 
ranges over all compositions of O is closed under conjugation and 
taking intersections. These subgroups are precisely the stabilizers 
which arise in the symmetric powers S^nO of O. Each subgroup H of G 
is contained in a unique minimal subgroup G_C.

Proof. The conjugate of G_C by g is G_{gC} .
The intersection of G_C and G_D is G_{C\cap D}.
We may identify S^dO as the set of formal symbols o_1^{a_1}...o_n^{a_n} 
where O = {o_1, ... ,o_n} and the a_i are non-negative integers which 
sum to d. For each r let O_r = {o_i | a_i=r}. This gives a composition 
C of O and the stabilizer of the product symbol as an element of S^d is 
G_C.
Given a subgroup H of G, Let O = O_1 \cup ... \cup O_t  be the 
orbits of H on O. This gives a composition C of O. Now H is a 
subgroup of G_C, and this is the unique minimal such subgroup of G 
which contains H.
QED

Corollary. Let O be a G-set. The G-sets G/G_C span a subring of the 
Burnside ring.

Proof This follows from closure under conjugation and intersections. QED

Corollary. Let O be a G-set. If K is a subgroup of G which is not of 
the form G_C then G/K does not appear as a coefficient in the Poincare 
series \sum S^d(O) t^d .

Proof This comes from the identification of the stabilizers in the 
symmetric powers as subgroups G_C.  QED
One may also approach this in a more complicated fashion from my formula 
for this series in terms of the Moebius function. One uses the fact that 
each subgroup H is contained in a unique minimal subgroup G_C and for all 
subgroups K lying between H and this G_C the function f_K is constant. 
Now a defining property of the Moebius function gives that the sum in my 
formula taken over those subgroups K is zero if H is not G_C. One can now 
proceed in an induction fashion to complete the argument.

The upshot of this is that the above subring of the Burnside ring is the 
one in which the coefficients of the Poincare series lie. To proceed 
further with the question of showing that the reciprocal of the Poincare 
series is a polynomial with coefficients in the Burnside ring (which for 
all I know might be true in the generality I have just outlined, but I 
really have no idea) I am interested to know firstly the structure of this 
subring e.g. in terms of the poset of the subgroups G_C. This in itself 
might be quite an interesting problem. After that it is a question of 
getting an appropriate formula for the (reciprocal of the) Poincare 
series, 
and ... who knows?

Peter Webb