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.S_{1};S_{2}/^{^}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 F_{1} andF_{2} be saturated fusion systems over the finite p–groups
S1 and S2, respectively, and let !1 and!2 be characteristic idempotents ofF_{1} and
F_{2}, respectively. If the .S1;S2/–pairs .P; / and .Q; / are .F_{1};F_{2}/–conjugate,
then

!2ıŒQ; ı!1D!2ıŒP; ı!1:

Proof Since !1 is left F_{1}–stable and !2 is right F_{2}–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 F_{1} andF_{2} be saturated fusion systems over the finite p–groups
S1 andS2, respectively and let!1 and !2 be characteristic idempotents ofF_{1} and
F_{2}, respectively. LetI be the set of .F_{1};F_{2}/–conjugacy classes of .S_{1};S_{2}/–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.S_{1};S_{2}/^{^}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;

whereas

(6) .M .ŒPi; ^{i}//p jS_{1}=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

(8) X

i2I

ci.!2ıŒPi; ^{i}ı!1/D0:
Put

I^{0}D fi 2I jci¤0g:

If I^{0} is nonempty, then let j be a maximal element ofI^{0} regarded as a poset under
.F_{1};F_{2}/–subconjugacy. By(7)there is a .S_{1};S_{2}/–pair.Q; /.Pj; j/ such that

_{Œ}Q; !2ıŒPj; ^{j}ı!1

¤0:

On the other hand, for i2I^{0}n fjg the maximality of j implies that ŒQ; is not
.F_{1};F_{2}/–subconjugate to.Pi; i/. Hence

_{Œ}Q;.M .-ŒPi; i//D0
and in particular

_{Œ}Q;.!2ıŒPi; iı!1/D0:
Now we get

_{Œ}Q;

X

i2I

ci.!2ıŒPi; iı!1/

! DX

i2I

ci_{Œ}Q;.!2ıŒPi; iı!1/

D X

i2InI^{0}

ci

„ƒ‚…

D0

_{Œ}Q;.!2ıŒPi; ^{i}ı!1/

C X

i2I^{0}nfjg

c_{i}_{Œ}Q;.!2ıŒP_{i}; iı!1/

„ ƒ‚ …

D0

C cj

„ƒ‚…

¤0

_{Œ}Q; !2ıŒPj; jı!1

„ ƒ‚ …

¤0

¤0;

contradicting(8). Therefore I^{0} must be empty and we conclude that the collection is
linearly independent.

Remark 5.3 Since the multiplicative identity ŒS_{1};i d of A.S_{1};S_{1}/^{^}p is a
characteris-tic idempotent forF_{S}_{1} andŒS_{2};i dis a characteristic idempotent forF_{S}_{2}, Propositions
5.1and5.2can also be used to obtain a basis for!2ıA.S_{1};S_{2}/^{^}p andA.S_{1};S_{2}/^{^}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

A_{F}.S;S/^{^}p !!A_{F}.S;S/^{^}p;

given by left multiplication by !. This projection can be easily described by

D X

ŒP;'-ŒS;id

_{Œ}P;'./ŒP; '7 !!ıD X

ŒP-ŒS

^{F}_{Œ}_{P}_{}./.!ıŒP; P/;

where the homomorphisms ^{F}_{Œ}_{P}_{} 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 ^{F}_{Œ}_{P}_{} and ^{Œ}_{F}^{P}^{} be the homomorphisms
A.S;S/^{^}p!Z^{^}p given by

^{F}_{Œ}_{P}_{}D X

ŒP;'

.FS;F/ŒP;P

_{Œ}P;'

and

^{Œ}F^{P}^{}D X

ŒQ;'

.F;FS/ŒP;^{P}

_{Œ}Q;':

Note that similarly we have

ı!D X

ŒP-ŒS

^{Œ}_{F}^{P}^{}./.ŒP; Pı!/:

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

!D!^{}C!^{};
where

!^{}D 1

jOutF.S/j X

'2OutF.S/

ŒS; '2M .ŒS;i d;F;F/

and

!^{}2M.ŒS;i d;F;F/ :
Furthermore,

^{F}_{Œ}_{P}_{}.!/D0
and

^{Œ}F^{P}^{}.!/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

!D!^{}C!^{};
where

!^{}2M .ŒS;i d/

and

!^{}2M .ŒS;i d/ :

From Properties (a) and (b) of characteristic idempotents, one deduces by standard
techniques that !^{} must be of the form

!^{}D˛ X

'2OutF.S/

ŒS; ';

where ˛2Z^{^}p. Now,

!^{}C!^{}D!D!^{2}D.!^{}/^{2}C.!^{}ı!^{}C!^{}ı!^{}C.!^{}/^{2}/;

where .!^{}/^{2}2M.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

.!^{}/^{2}D˛ jOutF.S/j !^{};
this implies that

˛2 f0;jOutF.S/j ^{1}g:

(Note that jOutF.S/j is prime to p since F is saturated.) By Property (c) of characteristic idempotents we have

1D.!/D.!^{}/C.!^{}/:

As a special case of(6)we have .M.ŒS;i d//p_{Z}^{^}_{p}, and it follows that

˛ jOutF.S/j D.!^{}/.!/1.mod p/:

Hence˛¤0, leaving only the possibility

!^{}D 1

jOutF.S/j X

'2Out^{F}.S/

ŒS; ':

To prove the second claim we start by deducing from Property (b) and the description
of!^{} above, that

!ı!^{}D!:

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

!^{}D X

ŒP;'ŒS;id

_{Œ}P;'.!/ŒP; ';

so

0D!ı!^{}D X

ŒPŒS

^{F}_{Œ}_{P}_{}.!/.!ıŒP; P/ ;

and the result for^{F}_{Œ}_{P}_{}.!/ follows upon noting that the collection
f!ıŒP; PjŒPŒSg

is linearly independent over Z^{^}_{p} byProposition 5.2andRemark 5.3.

The result for^{Œ}_{F}^{P}^{}.!/ 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

!D X

ŒP;'-ŒS;id

_{Œ}P;'.!/ŒP; ';

the method of proof is to set up a system of linear equations for the coefficients
_{Œ}P;'.!/, 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 'i0DP_{i}. 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;m_{0}/.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

^{j}_{i} WD^{'}_{P}^{i}_{i}^{j}
and

cijWD_{Œ}_{P}

i;'^{j}i.!/

for short. We will also use the shorthand notation P

i;j for the double sum Pn

iD0

Pm_{i}

jD0: Using this notation we can write

!DX

i;j

cijŒPi; 'ij:

Note that byLemma 2.6, Property (b1) implies that
j!^{}^{j}^{i}j D j!^{}^{0}^{i}j
for all pairs .i;j/. This can be rewritten as the equation

X

k;l

c_{kl}.jŒP_{k}; 'kl^{}^{j}^{i}j jŒP_{k}; 'kl^{}^{0}^{i}j/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

m0

X

jD0

c0jD1;

and for iD1; : : : ;n, we let Equation .i;0/ be the equation

mi

X

jD0

cijD0:

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

AcDb;

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; ^{}^{}^{R}j D j_{Q}nNSS.^{}_{R}; _{Q}/j DjNSS.^{}_{R}; _{Q}/j
jQj
for .S;S/–pairs .Q; / and .R; /, we see byRemark 3.4that

jŒQ; ^{}^{}^{R}j ¤0
if and only if

ŒR;

-.S;S/ŒQ; :

The chosen order of the basis ofA_{F}.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:

!F 2A.S;S/_{.}p/DA.S;S/˝Z_{.}p/:

(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.