The Hecke algebra of a Frobenius P-category

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by  Lluis Puig

The Hecke algebra
of a Frobenius P-category
Lluis Puig
CNRS,InstitutdeMath´ematiquesdeJussieu
6AvBizet,94340Joinville-le-Pont,France
1
1 [email protected]
0
1. Introduction
2

n
1.1. LetpbeaprimenumberandP afinitep-group;asusual,denoteby
a
F theFrobeniuscategory ofP [4,I1],namelythecategorywheretheobjects
J P
are the subgroups of P and the morphisms are the group homomorphisms
6
between them induced by the inner automorphisms of P . Recall that a divi-

] sible P-category F [4,2.3]isasubcategoryofthecategory of finitegroups Gr
R
such that it contains F and has the same objects, that all the morphisms
P
G
are injective group homomorphisms and that, for any subgroup Q of P , the
. category F of F-morphisms to Q is a full subcategory of Gr [4, 1.7].
h |Q |Q
t
a 1.2. Moreprecisely,denotingbyF(Q)thegroupofF-automorphismsof
m anysubgroupQofP ,recallthatthenF isaFrobenius P-category [4,Propo-
[ sition 2.11] if FP(P) is a Sylow p-subgroup of F(P) and if any F-morphism
ϕ:Q→P fulfilling
1
v
ζ C ϕ(Q) =C (ζ◦ϕ)(Q) 1.2.1
8 P P
(cid:16) (cid:0) (cid:1)(cid:17) (cid:0) (cid:1)
8
foranyF-morphismζ:ϕ(Q)·C ϕ(Q) →P,canbeextendedtoanF-mor-
2 P
1 phismfromtheconverseimagein(cid:0)P of(cid:1)theintersectionF (Q)∩ϕ∗F ϕ(Q) ,
P P
1. where we denote by ϕ∗:ϕ(Q) ∼= Q the inverse of the isomorphism(cid:0)induce(cid:1)d
0 by ϕ, and by ϕ∗F ϕ(Q) the correspondingimage of F ϕ(Q) in Aut(Q).
P P
1
(cid:0) (cid:1) (cid:0) (cid:1)
1 1.3. On the one hand, in a Frobenius P-category F still holds a gene-
: ral form for the Alperin Fusion Theorem which, in some sense, provides
v
i a “decomposition” of the F-morphisms [4, Corollary 5.14]. On the other
X hand, restricting ourselves to the full subcategory Fsc over the subgroups
r
Q of P fulfilling C ϕ(Q) ⊂ ϕ(Q) for any ϕ ∈ F(P,Q), in the quotient
a P
categoryF˜sc ofFsc b(cid:0)ythei(cid:1)nnerautomorphismsoftheobjectsthemorphisms
admit a “canonical decomposition” in terms of suitable subsets F˜sc(Q,R)
˜ιP
R
of F˜sc-morphisms as it follows from [4, Proposition 6.7]. An old aim we had
was to find a common framework where we could formulate both facts.
1.4. More recently, we have proved that the Frobenius P-categories are
justanavatar ofthebasic P×P-sets [4,21.4]—namelythefinitenonempty
P ×P-sets Ω such that {1}×P acts freely on Ω, we have
Ω◦ ∼=Ω and |Ω|/|P|6≡0modp 1.4.1
2 Lluis Puig
where we denote by Ω◦ the P ×P-set obtainedexchangingboth factorsand,
for any subgroup Q of P and any group homomorphism ϕ:Q → P such
that Ω contains a P ×P-subset isomorphic to (P ×P)/∆ (Q), we have a
ϕ
Q×P-isomorphism
Res (Ω)∼=Res (Ω) 1.4.2,
ϕ×idP ιPQ×idP
where we set ∆ (Q) = {(ϕ(u),u)} and denote by ιP the corresponding
ϕ u∈Q Q
inclusion map. The last but not the least, their generalization by Kari Rag-
narsson and Radu Stancu to virtual P ×P-sets [6] has driven our attention
upon the so-called double Burnside ring of P.
1.5. Inthispaperweintroduceanewavatar ofaFrobeniusP-categoryF
in the form of a suitable subring H of the double Burnside ring of P —
F
called the Hecke algebra of F — where we are able to formulate all the
results mentioned above, in some sense providing the framework we were
looking for. The choice of the name comes from the case where F is the
Frobenius category of a finite Chevalley group in the defining characteristic
or, more generally, of a finite group G, where an analogous definition of the
Hecke algebra is usually considered; in this case, the Hecke algebra of G is
isomorphic to a quotient of a suitable extension of H (see §8 below).
F
2. The quoted framework
2.1. Let us denote by P-Set the category of finite P-sets endowed with
the disjoint union and with the inner direct product mapping any pair of
finite P-sets X and Y on the P-set — still noted X ×Y — obtained from
the P ×P-set X ×Y restricted through the diagonal map ∆:P → P ×P ;
note that, for a third finite P-set Z, we have a canonical bijection
Hom (Z,X ×Y)∼=Hom (Z,X)×Hom (Z,Y) 2.1.1.
P P P
Then, we denote by Fct(P-Set) the categoryof functors from P-Set to itself
preserving the disjoint unions, endowed with the disjoint union induced by
the disjoint union in P-Set and with the composition of functors.
2.2. More generally, if P′ is a second finite p-group, we often have to
consider the category Fct(P-Set,P′-Set) of functors from P-Set to P′-Set
preserving the disjoint union, which is also endowed with a disjoint union;
let us say that a functor f:P-Set → P′-Set is indecomposable if it is not
the disjoint union of two nonempty functors in Fct(P-Set,P′-Set). If P′′ is
a third finite p-group, we clearly have a distributivity, namely
(g⊔g′)◦f=(g◦f)⊔(g′◦f) and g◦(f⊔f′)=(g◦f)⊔(g◦f′) 2.2.1
for any functors f and f′ in Fct(P-Set,P′-Set) and any functors g and g′ in
Fct(P′-Set,P′′-Set). We denote by Nat(f,f′) the set of natural maps from f
Lluis Puig 3
to f′, but write Nat(f) insteadof Nat(f,f); it is clearthat for a third functor
f′′:P-Set→P′-Set we have
Nat(f⊔f′,f′′)∼=Nat(f,f′′)×Nat(f′,f′′) 2.2.2.
2.3. On the other hand, if α:P′ → P is a group homomorphism, the
restriction defines an evident functor
res :P-Set−→P′-Set 2.3.1
α
which admits a left adjoin functor
ind :P′-Set−→P-Set 2.3.2;
α
hence,foranyfunctorsginFct(P′′-Set,P′-Set)andhinFct(P′′-Set,P-Set)
we have a canonical bijection
Nat(ind ◦g,h)∼=Nat(g,res ◦h) 2.3.3.
α α
2.4. Notethat,anypairoffunctorsfandf′ inFct(P-Set,P′-Set)defines
a functor
f×f′ :P-Set×P-Set−→P′-Set 2.4.1
mapping anypair offinite P-sets X andX′ onthe P′-setf(X)×f′(X′), and
any pair of P-set maps a:X → Y and a′:X′ → Y′ on the evident P′-set
map
f(a)×f′(a′):f(X)×f′(X′)−→f(Y)×f′(Y′) 2.4.2;
then, for any pair of functors g and g′ in Fct(P-Set,P′-Set) we have an
evident injection
Nat(g,f)×Nat(g′,f′)−→Nat(g×g′,f×f′) 2.4.3.
whichisbijectiveifgandg′bothmapthetrivial P-Setonthetrivial P′-Set.
Indeed, according to the bijection 2.1.1, any P-set map
g(X)×g′(X′)−→f(X)×f′(X′) 2.4.4
determines and is determined by the corresponding two P-set maps
g(X)×g′(X′)−→f(X) and g(X)×g′(X′)−→f′(X′) 2.4.5;
but, it follows from the naturalness that the left-hand map in 2.4.5 is deter-
minedbythemapwhichcorrespondstothecasewhereX′isthetrivial P-set
andthenitdeterminesanelementofNat(g,f);similarly,theright-handmap
in 2.4.5 determines an element of Nat(g′,f′); finally, it is easily checked that
this correspondence is the inverse of the injection 2.4.3.
4 Lluis Puig
2.5. Actually, we are only interested in the full subcategory BP×P′
ofFct(P-Set,P′-Set)overthefunctorsdeterminedbyP′×P-sets;explicitly,
a P′×P-set Ω determines a functor f mapping any P-set X on the P′-set
Ω
— noted Ω× X — formed by the set of orbits of P on Ω×X through the
P
actionsending(ω,x)∈Ω×X to (1,u)·ω,u·x foranyu∈P ,endowedwith
the action of P′ mapping the orb(cid:0)it of (ω,x) o(cid:1)n the orbit of (u′,1)·ω,x for
any u′ ∈P — we often write u′·ω and ω·u−1 instead of(u′,1(cid:0))·ω and (1,(cid:1)u)·ω
respectively; it is clear that f preserves the disjoint unions. For instance,
Ω
for P′′ =P , the restriction and the induction functors above
res :P-Set−→P′-Set and ind :P′-Set−→P-Set 2.5.1
α α
are respectively determined by the P′×P- and the P ×P′-sets P and P
α α
denoting the respective P′ ×P- and P ×P′-set structures over P sending
v ∈ P to α(u′)vu−1 and to uvα(u′)−1 for any u∈ P and any u′ ∈ P′. Note
thatthefunctorf determinestheP×P-setΩ;indeed,consideringtheP-set
Ω
P and its group of automorphisms in P-Set — which is isomorphic to P —
we get a P′×P-set structure on f (P) and, with this structure, it is clear
Ω
that f (P) is isomorphic to Ω. In particular, for any P′×P-set Ω′, we have
Ω
a canonical bijection
Nat(fΩ,fΩ′)∼=HomP′×P(Ω,Ω′) 2.5.2.
2.6. It is clear that BP×P′ is closed with respect to the disjoint union,
namely we have fΩ⊔fΩ′ = fΩ⊔Ω′, and an indecomposable functor in BP×P′
is given by the transitive P′ × P-set (P′ × P)/D where D is a subgroup
of P′×P ; shortly we set
fD =f(P′×P)/D 2.6.1.
Then, foranyP′×P-setΩ,it followsfromisomorphism2.5.2thata natural
map µ:f →f is induced by a P′×P-set map (cf. 2.5.2)
D Ω
m :(P′×P)/D−→Ω 2.6.2
µ
which is actually determined by the image in Ω of the class in (P′×P)/D
of (1,1), and this image clearly belongs to the set of D-fixed elements ΩD;
conversely,anyelementωofΩD determinesaP′×P-setmapfrom(P′×P)/D
to Ω sending the class of (1,1) to ω, and therefore it determines a natural
map from f to f ; thus, we have a bijection
D Ω
Nat(f ,f )∼=ΩD 2.6.3.
D Ω
2.7. In particular, it is clear that the group
Z(fD)=CP′×P(D) Z(D) 2.7.1
(cid:14)
acts on the set of natural maps Nat(f ,f ). Actually, all these subgroups
D Ω
Z(f ) ⊂ Nat(f ), where D runs over the set of subgroups of P′ × P ,
D D
Lluis Puig 5
◦
determine an interior structure in (BP×P′) [4, 1.3] and, coherently, for an-
other subgroup E of P′×P , we set
Nat(f ,f )=Z(f ) Nat(f ,f ) 2.7.2.
D E D D E
(cid:15)
g
Note that, for any pair of functors g and g′ in BP×P′ we get
Nat(f ,g⊔g′)∼=Nat(f ,g)⊔Nat(f ,g′) 2.7.3.
D D D
2.8. Moreover, B is also closed with respect to the composition,
P×P
namely we have
fΩ◦fΩ′ =fΩ×PΩ′ 2.8.1
foranyP×P-setsΩandΩ′.Ontheotherhand,denotingbyΩ◦ theopposite
◦
P ×P-set of a P ×P-set Ω, the correspondence mapping fΩ on (fΩ) =fΩ◦
definesanauto-equivalencetofB whichpreservesthedisjointunionand
P×P
reverses the composition; then, it follows from bijection 2.5.2 that, for any
pair of functors f and g in B , we still have a canonical bijection
P×P
Nat(f◦,g◦)∼=Nat(f,g) 2.8.2.
Let us call symmetric any t-stable subcategory of B .
P×P
2.9. A particular kind of functors in B is provided by the inner
P×P
directproductinP-Set;indeed,foranyP-setX,thecorrespondencesending
any P-set Y to the P-set X×Y and any P-set map g:Y →Y′ to the P-set
map
id ×g :X×Y −→X ×Y′ 2.9.1
X
definesafunctorm fromP-SettoP-Setpreservingthedisjointunions,and
X
it is easily checked that, for any subgroup Q of P, the functor determined
by the P-set P/Q coincides with the functor determined by the P ×P-set
P × P; more precisely, we have a functor
Q
mP :P-Set−→B 2.9.2
P
mapping any P-set X on m and any P-set map f:X →X′ to the natural
X
map µf:mX →mX′ sending Y to the P-set map
f ×id :X ×Y −→X′×Y 2.9.3.
Y
Note that mP preserves the disjoint unions, maps the inner direct prod-
uct in P-Set on the composition in B , and is invariant by the auto-
P×P
equivalence t of B above.
P×P
6 Lluis Puig
2.10. Actually, P hasanobviousP×P-setstructureanditis clearthat
f is the identity functor. More generally, for any subgroup Q of P and any
P
group homomorphism ϕ:Q→P, as in 1.4 above we set
∆ (Q)= ϕ(u),u } 2.10.1
ϕ u∈Q
(cid:8)(cid:0) (cid:1)
and, coherently with our notation in 2.5, we still set
P × P =(P ×P)/∆ (Q) and f =f =ind ◦res 2.10.2;
ϕ Q ϕ Q,ϕ Pϕ×QP ϕ ιPQ
itisquiteclearthatthis functorpreservestheprojective objectsinP-Setor,
equivalently, the P-sets where P acts freely; conversely, any indecomposable
functor in B which preserves the projective P-sets has this form.
P
2.11. More explicitly, any functor f in B preserving the projective
P×P
P-sets has the form f = f for a suitable finite family {Q ,ϕ }
i∈I Qi,ϕi i i i∈I
where Qi are subgroups oFf P and ϕi:Qi → P are group homomorphisms;
then, for another functor g in B preserving the projective P-sets, given
P
by the finite family {R ,ψ } , there is a natural map µ:g → f if and
j j j∈J
only if there are a map m:J → I and, for any j ∈ J, a natural map
µ :f →f or, equivalently, a P ×P-set map
j Rj,ψj Qm(j),ϕm(j)
m :P × P −→P × P 2.11.1,
j ψj Rj ϕm(j) Qm(j)
which amounts to saying that there are u ,v ∈P such that
j j
R ⊂(Q )uj and ψ =κ ◦ϕ ◦κ 2.11.2
j m(j) j vj m(j) uj
where κ :R →Q andκ :P →P respectively denote the conjugation
uj j m(j) vj
by u and by v .
j j
2.12. In particular, for any indecomposable functors f and f in
Q,ϕ R,ψ
◦ ◦
B such that (f ) and (f ) still preserve the projective P-sets, de-
P×P Q,ϕ R,ψ
noting by ϕ :Q∼=ϕ(Q) and ψ :R∼=ψ(R) the corresponding isomorphisms
∗ ∗
and setting ϕ∗ = (ϕ )−1 and ψ∗ = (ψ )−1, it follows from condition 2.11.2
∗ ∗
above that any natural map from f to f determines an element of the
R,ψ Q,ϕ
intersection
F˜ (Q,R)∩ ϕ˜∗◦F˜ ϕ(Q),ψ(R) ◦ψ˜ 2.12.1,
P P ∗
(cid:0) (cid:0) (cid:1) (cid:1)
namely an element κ˜ in F˜ (Q,R) such that the composition ϕ˜ ◦κ˜ ◦ψ˜∗
u P ∗ u
belongs to F˜ ϕ(Q),ψ(R) ; moreover, it is easily checked that this cor-
P
respondence de(cid:0)termines a b(cid:1)ijection
Nat(f ,f )∼=F˜ (Q,R)∩ ϕ˜∗◦F˜ ϕ(Q),ψ(R) ◦ψ˜ 2.12.2.
R,ψ Q,ϕ P P ∗
(cid:0) (cid:0) (cid:1) (cid:1)
g
Lluis Puig 7
2.13. Moreover,foranypairofsubgroupsQandRofP ,andanypairof
grouphomomorphismsϕ andψ fromQ andR to P respectively,the functor
f ◦f is determined by the P ×P-set
Q,ϕ R,ψ
(P × P)× (P × P)=P × P × (P × P)
ϕ Q P ψ R ϕ Q P ψ R
(cid:0) (cid:1)
=P × (P × P )× P 2.13.1;
ϕ Q P ψ R
(cid:0) (cid:1)
=P × (P × P)
ϕ Q ψ R
but, choosing a set of representatives W ⊂ P for the set of double classes
Q\P/ψ(R), we have the disjoint decomposition
P = Q·w·ψ(R) 2.13.2;
wG∈W
moreover,foranyw ∈W ,itisclearthatQ·w·ψ(R)hasaQ×R-setstructure
sendinguwψ(v)−1 tou′uwψ(v′v)−1 foranyu,u′ ∈Qandanyv,v′ ∈R;more
precisely, setting U =ψ−1 Qw∩ψ(R) and denoting by ψ :U →Qw the
w w w
restriction of ψ and by κ :(cid:0)Qw →Q the(cid:1)corresponding conjugation by w, it
w
is clear that we have a Q×R-set isomorphism
Q·w·ψ(R)∼=Q × R 2.13.3;
κw◦ψw Uw
finally, we get a P ×P-set isomorphism
(P × P)× (P × P)∼= P × (Q × R)× P
ϕ Q P ψ R ϕ Q κw◦ψw Uw R
wG∈W
2.13.4.
∼= P × P
ϕ◦κw◦ψw Uw
wG∈W
3. A version for divisible and Frobenius P-categories
3.1. For any divisible P-category F, let us denote by H the full sub-
F
category of B over the functors which are isomorphic to disjoint unions
P×P
offunctorsf whereQisasubgroupofP andϕ:Q→P isanF-morphism;
Q,ϕ
thus, H is closed with respect to the disjoint unions, its elements preserve
F
the projective P-sets, and is clearly symmetric (cf. 2.8). Moreover, H is
F
closedwithrespecttothecomposition;indeed,forasecondsubgroupRofP
and any F-morphism ψ:R→P, with the notation in 2.13 above we have
f ◦f = f 3.1.1
Q,ϕ R,ψ Uw,ϕ◦κw◦ψw
wG∈W
and the divisibility in F implies that, for any w ∈ W, ψ and κ are
w w
F-morphisms and therefore ϕ ◦ κ ◦ ψ is also an F-morphism. In par-
w w
ticular, since F contains the Frobenius category F of P , H contains H
P F FP
— for short, we write H instead of H ; note that H coincides with the
P FP P
image of the functor mP .
8 Lluis Puig
3.2. More generally, the divisibility of F implies that
3.2.1 Any functor f in B admitting a natural map to a functor in H
P×P F
is an H -object.
F
Indeed, we actually may assume that f = f where D is a subgroup of
D
P ×P; then, our hypothesis and 2.11 above imply that there are a functor
f , where Q is a subgroup of P and ϕ:Q → P is an F-morphism, and a
Q,ϕ
natural map µ:f →f , which amounts to saying that there are u,v ∈P
D Q,ϕ
such that
D ⊂∆ (Q)(u,v) 3.2.2;
ϕ
since F is divisible and contains F , it easily follows that D =∆ (R) for a
P ψ
subgroup R of Qv and an F-morphism ψ:R → P. Actually, this condition
is equivalent to the divisibility of F as our next result shows.
Proposition 3.3. Let H be a full symmetric subcategory of B con-
P×P
taining H , where all the elements preserve the projective P-sets, closed with
P
respect to disjoint unions and tocomposition, and fulfilling the following con-
dition:
3.3.1. Any functor f in B admitting a natural map to a functor in H is
P×P
an H-object.
Then, the family of sets
F (P,Q)={ϕ∈Hom(Q,P)|f is an object of H} 3.3.2,
H Q,ϕ
where Q runs over the set of subgroups of P , determine a divisible P-cate-
gory F .
H
Proof: Since H contains H , it is clear that F (P,Q) contains F (P,Q);
P H P
moreover,following [1, 2.4.1], let Q and R be subgroups of P and consider a
group homomorphism θ:R→Q such that
F (P,R)∩ F (P,Q)◦θ 6=∅ 3.3.3;
H H
(cid:0) (cid:1)
thatistosay,weassumethatthereisagrouphomomorphismϕ:Q→P such
◦
that f and f are H-objects; hence, the composition (f ) ◦f is
Q,ϕ R,ϕ◦θ Q,ϕ R,ϕ◦θ
an H-object too; but, it is clear that
◦
(f ) =f 3.3.4
Q,ϕ ϕ(Q),ιP◦ϕ∗
Q
where, as in 1.2 above, ϕ∗ denotes the inverse of the isomorphismQ∼=ϕ(Q)
determined by ϕ, and then it follows from equality 3.1.1 that we have a
natural map
◦
f −→(f ) ◦f 3.3.5;
R,ιP◦θ Q,ϕ R,ϕ◦θ
Q
consequently, f is also an H-object.
R,ιP◦θ
Q
Lluis Puig 9
Now,sinceforanyϕ′ ∈FH(P,Q)thefunctorfQ,ϕ′◦fR,ιP◦θ isanH-object
Q
and it follows again from equality 3.1.1 that we have a natural map
fR,ϕ′◦θ −→fQ,ϕ′ ◦fR,ιP◦θ 3.3.6,
Q
fR,ϕ′◦θ is an H-object too; in conclusion, we get
F (P,Q)◦θ ⊂F (P,R) 3.3.7,
H H
so that the proposition follows from [1, 2.4.1].
3.4. Let us call divisible H -category any full symmetric subcategory
P
H of B containing H , where all the elements preserve the projective
P×P P
P-sets, closed with respect to disjoint unions and to composition, and ful-
filling condition 3.3.1; then, the proposition above guarantees that H deter-
mines a divisible P-category F and we already have seen that any divisible
H
P-category F determines a divisible H -category H . Moreover, it is quite
P F
clear that
H =H and F =F 3.4.1,
FH HF
so that these correspondencesdetermine a bijection betweenthe sets of divi-
sible P-categories anddivisible H -categories; wediscribebelowtheimageof
P
theFrobeniusP-categories. RecallthatwedenotebyF˜ thequotientcategory
of F by the inner automorphisms of the objects [4, 1.3].
3.5. Firstofall,foranyfunctorfinB whichpreservestheprojective
P×P
P-sets we define a positive integer ℓ(f) by
|f(P)|=ℓ(f)|P| 3.5.1;
for another such a functor g in B , we have
P×P
ℓ(f⊔g)=ℓ(f)+ℓ(g) and ℓ(f◦g)=ℓ(f)ℓ(g) 3.5.2.
Then, for any divisible H -category H denote by
P
ℓ :H−→N 3.5.3
H
the restrictionto H ofthis correspondenceandby Ker(ℓ ) the setof isomor-
H
phism classes of functors f in H fulfilling ℓ(f) = 1; it is quite clear that the
composition in H induces a groupstructure in Ker(ℓ ) and we actually have
H
Ker(ℓ )∼=F˜ (P) 3.5.4.
H H
3.6. On the other hand, accordingto 2.6 above,for any indecomposable
functor f in B , a natural map from f to f is determined by the
D P×P D D
image (u,v) ∈ (P × P)/D of (1,1) and then it is easily checked that the
element(u,v)∈P×P normalizesthe subgroupD ofP×P ;conversely,any
(u,v)∈N (D) determines a P ×P-set automorphism of (P ×P)/D and
P×P
therefore it determines a natural automorphism f ∼=f , so that we have
D D
Nat(f )∼=N¯ (D) 3.6.1
D P×P
10 Lluis Puig
and Nat(f ) acts on (P × P)/D. We are specially interested in the case
D
where f preserves the projective P-sets or, equivalently, where D =∆ (Q)
D ϕ
for a subgroup Q of P and a group homomorphism ϕ:Q → P ; then, from
bijection 2.12.2, we get the exact sequence (cf. 2.7)
1−→Z(f )−→Nat(f )−→F˜ (Q)∩ϕ∗F˜ ϕ(Q) −→1 3.6.2.
Q,ϕ Q,ϕ P P
(cid:0) (cid:1)
Proposition 3.7. Let H be a divisible H -category. Then F is a Frobenius
P H
P-category if and only if Ker(ℓ ) is a p′-group and, for any indecomposable
H
functor f in H such that we have
Q,ϕ
|Z(fQ,ϕ′)|≤|Z(fQ,ϕ)| 3.7.1
for any indecomposable functor fQ,ϕ′ in H, therearean indecomposable func-
tor f in H and a natural map µ from f to f such that the stabilizer
R,ψ Q,ϕ R,ψ
of µ∈Nat(f ,f ) in Nat(f ) is a complement of the image of Z(f ).
Q,ϕ R,ψ Q,ϕ Q,ϕ
Proof: BytheverydefinitionsofZ(f )andofF ,notethatthecondition
Q,ϕ H
on f is equivalent to
Q,ϕ
C ϕ′(Q) ≤ C ϕ(Q) 3.7.2
P P
(cid:12) (cid:0) (cid:1)(cid:12) (cid:12) (cid:0) (cid:1)(cid:12)
for any ϕ′ ∈ F (P,Q) (cid:12)which impli(cid:12)es t(cid:12)hat ϕ(Q) i(cid:12)s fully centralized in F
H H
[4, Proposition 2.7]. If F is a Frobenius P-category then, denoting by R
H
the converse image in P of F˜ (Q)∩ϕ∗F˜ ϕ(Q) , we know that there exists
P P
ψ ∈F (P,R)extendingϕor,equivalently,(cid:0)sucht(cid:1)hat∆ (R)contains∆ (Q);
H ψ ϕ
in this case, it is clear that ∆ (Q) is actually normal in ∆ (R), so that we
ϕ ψ
get
N ∆ (Q) = C (Q)×C ϕ(Q) ·∆ (R) 3.7.3,
P×P ϕ P P ψ
(cid:0) (cid:1) (cid:16) (cid:0) (cid:1)(cid:17)
and therefore the exact sequence 3.6.2 above is split; moreover, we have an
evident natural map µ:f →f and the stabilizer of µ∈Nat(f ,f )
Q,ϕ R,ψ Q,ϕ R,ψ
in Nat(f ) is a complement of the image of Z(f ).
Q,ϕ Q,ϕ
Conversely, assume that the condition on f holds and that there
Q,ϕ
are an indecomposable functor f in H and a natural map ν from f
T,η Q,ϕ
to f such that the stabilizer of ν ∈ Nat(f ,f ) in Nat(f ) is a com-
T,η Q,ϕ T,η Q,ϕ
plement of the image of Z(f ); then, with the notation above, we claim
Q,ϕ
that there exists ψ ∈F (P,R) extending ϕ. Indeed, it is clear that ν is de-
H
terminedby(u,v)∈T ∆ (Q),∆ (T) andthereforethe converseimage
P×P ϕ η
in N ∆ (Q) of the s(cid:0)tabilizer of ν c(cid:1)oincides with N ∆ (Q)
P×P ϕ ∆η(T)(u,v) ϕ
and cover(cid:0)s F˜ (Q(cid:1))∩ϕ∗F˜ ϕ(Q) , which forces (cid:0) (cid:1)
P P
(cid:0) (cid:1)
N ∆ (Q) =∆ (R)⊃∆ (Q) 3.7.4
∆η(T)(u,v) ϕ ψ ϕ
(cid:0) (cid:1)
whereψ(w)=vη(uwu−1)v−1 foranyw ∈R;hence,ψextendsϕandbelongs
to F (P,R), proving the claim.
H