In this section we perform a further study of the characteristic idempotents introduced in the previous section. We discover that the effect of multiplicaton by a characteristic idempotent onA.S1;S2/^p essentially amounts to quotienting out fusion conjugacy in the source or target, as appropriate. This allows us to glean important information about the structure of a characteristic idempotent in the ŒP; '–basis, which allows us to prove its uniqueness, and will also prove surprisingly useful for proving later naturality results.
Proposition 5.1 Let F1 andF2 be saturated fusion systems over the finite p–groups S1 and S2, respectively, and let !1 and!2 be characteristic idempotents ofF1 and F2, respectively. If the .S1;S2/–pairs .P; / and .Q; / are .F1;F2/–conjugate, then
!2ıŒQ; ı!1D!2ıŒP; ı!1:
Proof Since !1 is left F1–stable and !2 is right F2–stable, this is a special case of Lemma 3.12.
Although the following proposition may be of limited interest in its own right, it is the central result of this paper.
Proposition 5.2 Let F1 andF2 be saturated fusion systems over the finite p–groups S1 andS2, respectively and let!1 and !2 be characteristic idempotents ofF1 and F2, respectively. LetI be the set of .F1;F2/–conjugacy classes of .S1;S2/–pairs, and pick a representative .Pi; i/ for eachi2I. Then the collection
f!2ıŒPi; iı!1ji2Ig forms aZ^p–basis for !2ıA.S1;S2/^pı!1.
Proof It follows fromProposition 5.1that !2ıA.S1;S2/^pı!1 is spanned by the collection, so it suffices to prove linear independence. By Property (b) of characteristic idempotents andLemma 3.8we have
!2ıŒPi; iı!12M.-ŒPi; i/
for eachi 2I. Note however, that
.!2ıŒPi; iı!1/D.!2/.ŒPi; i/.!1/D1 jS1=Pij 1D jS1=Pij;
(6) .M .ŒPi; i//p jS1=PijZ^p: Therefore,
(7) !2ıŒPi; iı!12M .-ŒPi; i/nM .ŒPi; i/
for eachi 2I.
Now, let ci2Z^p for eachi 2I and assume that
ci.!2ıŒPi; iı!1/D0: Put
I0D fi 2I jci¤0g:
If I0 is nonempty, then let j be a maximal element ofI0 regarded as a poset under .F1;F2/–subconjugacy. By(7)there is a .S1;S2/–pair.Q; /.Pj; j/ such that
ŒQ; !2ıŒPj; jı!1
On the other hand, for i2I0n fjg the maximality of j implies that ŒQ; is not .F1;F2/–subconjugate to.Pi; i/. Hence
ŒQ;.M .-ŒPi; i//D0 and in particular
ŒQ;.!2ıŒPi; iı!1/D0: Now we get
„ ƒ‚ …
ŒQ; !2ıŒPj; jı!1
„ ƒ‚ …
contradicting(8). Therefore I0 must be empty and we conclude that the collection is linearly independent.
Remark 5.3 Since the multiplicative identity ŒS1;i d of A.S1;S1/^p is a characteris-tic idempotent forFS1 andŒS2;i dis a characteristic idempotent forFS2, Propositions 5.1and5.2can also be used to obtain a basis for!2ıA.S1;S2/^p andA.S1;S2/^pı!1.
We determine the structure of a characteristic idempotent ! by carefully analyzing the idempotence relation!ı!D!. In light of the previous proposition, a convenient tool for doing this is the projection
given by left multiplication by !. This projection can be easily described by
ŒP;'./ŒP; '7 !!ıD X
where the homomorphisms FŒP are as in the following definition.
Definition 5.4 Let F be a fusion system over a finite p–group S. For each S– conjugacy class ŒP of subgroups of S, let FŒP and ŒFP be the homomorphisms A.S;S/^p!Z^p given by
Note that similarly we have
The following lemma now effectively allows us to determine the structure of character-istic idempotents.
Lemma 5.5 Let F be a saturated fusion system over a finitep–group S and let !be a characteristic idempotent ofF. Then
ŒS; '2M .ŒS;i d;F;F/
!2M.ŒS;i d;F;F/ : Furthermore,
ŒFP.!/D0 for all proper subgroupsP <S.
Proof There is a direct sum of modules
AF.S;S/^pDM.-ŒS;i d/DM.ŒS;i d/˚M .ŒS;i d/ ;
where M .ŒS;i d/ is a two-sided ideal of AF.S;S/^p (by Lemma 3.8), and M .ŒS;i d/is a subring ofAF.S;S/^p (as can be easily checked). We can therefore uniquely write
!2M .ŒS;i d/
!2M .ŒS;i d/ :
From Properties (a) and (b) of characteristic idempotents, one deduces by standard techniques that ! must be of the form
where ˛2Z^p. Now,
where .!/22M.ŒS;i d/ and the second term is in the idealM .ŒS;i d/. By uniqueness of such decompositions! is therefore an idempotent, and since
.!/2D˛ jOutF.S/j !; this implies that
˛2 f0;jOutF.S/j 1g:
(Note that jOutF.S/j is prime to p since F is saturated.) By Property (c) of characteristic idempotents we have
As a special case of(6)we have .M.ŒS;i d//pZ^p, and it follows that
˛ jOutF.S/j D.!/.!/1.mod p/:
Hence˛¤0, leaving only the possibility
To prove the second claim we start by deducing from Property (b) and the description of! above, that
Hence the idempotence of ! yields
!D!ı!D!ı.!C!/D!C!ı!; and we get
!ı!D0: By Property (a) andRemark 3.7, we can write
FŒP.!/.!ıŒP; P/ ;
and the result forFŒP.!/ follows upon noting that the collection f!ıŒP; PjŒPŒSg
is linearly independent over Z^p byProposition 5.2andRemark 5.3.
The result forŒFP.!/ is proved similarly by performing the analogous simplifications of!ı!D0. We omit the details.
The lemma has an interesting consequence.
Proposition 5.6 Every saturated fusion system has a unique characteristic idempotent.
Proof Let F be a saturated fusion system over a finitep–groupS. ByProposition 4.9,F has a characteristic idempotent !. We proceed to prove uniqueness. Recalling that we can write
the method of proof is to set up a system of linear equations for the coefficients ŒP;'.!/, and show that this system is fully determined and thus uniquely determines the coefficients. Note that we need not show that the system is solvable since the existence of a characteristic idempotent has already been established.
To present the argument it is helpful to index the basis ofAF.S;S/^p. Take representa-tivesP0;P1; : : : ;Pn for theS–conjugacy classes of subgroups of S, labelled in such a way that
jP0j jP1j : : : jPnj:
For eachi2 f0;1; : : : ;ng;pick representatives'i0; 'i1; : : : ; 'imi for theS–conjugacy classes of homomorphisms in HomF.P;S/, now labelled so that 'i0DPi. The collection fŒPi; 'ijj0in;0jmigis then a basis for AF.S;S/^p. We order the basis through the lexicographic order of the indexing set:
.0;0/.0;1/ .0;m0/.1;0/.1;1/ .n;mn/:
This ordering has the property thatŒPi; 'ij
-.S;S/ŒPk; 'klimplies .k;l/.i;j/:
For the remainder of this proof, a pair .i;j/ is understood to satisfy 0in and 1j mi. For such a pair .i;j/, write
ji WD'Piij and
for short. We will also use the shorthand notation P
i;j for the double sum Pn
jD0: Using this notation we can write
Note that byLemma 2.6, Property (b1) implies that j!jij D j!0ij for all pairs .i;j/. This can be rewritten as the equation
ckl.jŒPk; 'kljij jŒPk; 'kl0ij/D0;
which we refer to as Equation .i;j/whenj ¤0. WhenjD0, this equation becomes trivial. Instead we consider the equations given byLemma 5.5. That is, we let Equation .0;0/ be the equation
and for iD1; : : : ;n, we let Equation .i;0/ be the equation
If we now write Equations .0;0/ to.i;mi/one below the other going by the lexico-graphic order, we obtain a system of equations, which can be represented on matrix form as
where c is a vector with entries cij, b is a vector with 1 in its first entry and 0 everywhere else, while Ais a square matrix whose rows and columns are both indexed by pairs.i;j/. The proof is completed by showing that A has nonzero determinant.
There is an obvious way to regard A as a.nn/block matrix where the blocks are indexed by i. We show thatA is in fact a lower triangular block matrix. Since
jŒQ; Rj D jQnNSS.R; Q/j DjNSS.R; Q/j jQj for .S;S/–pairs .Q; / and .R; /, we see byRemark 3.4that
jŒQ; Rj ¤0 if and only if
The chosen order of the basis ofAF.S;S/^p therefore guarantees that A is a lower triangular block matrix.
It now suffices to show that the matrices occurring on the diagonal of A have nonzero determinant. The i-th matrix on the diagonal has the form
Ai D Direct calculation shows that the determinant of Ai is
det.Ai/D which is nonzero since all the aij’s are positive integers.
The previous proposition allows us to speak of the characteristic idempotent of a saturated fusion system.
Definition 5.7 For a saturated fusion system F, let !F denote the characteristic idempotent ofF.
Remark 5.8 We make the following observations about the proof ofProposition 5.6.
(1) As a byproduct of the proof we have produced an explicit system of equations which we can solve to produce !F.
(2) The coefficients in these equations are all integers. Therefore!F can be regarded as an element in thep–localized double Burnside ring:
(3) The proof actually shows that !F is the unique idempotent in A.S;S/^p (or A.S;S/.p/) with the Linckelmann–Webb Properties (a), (b1) and (c). A sym-metric argument shows that !F is the unique idempotent in A.S;S/^p with Properties (a), (b2) and (c). We are therefore in the surprising situation that for an idempotent inA.S;S/^p with Properties (a) and (c), the presence of either stability Property (b1) or (b2) implies the presence of the other.