Contents lists available atScienceDirect
Journal
of
Algebra
www.elsevier.com/locate/jalgebra
On
the
basis
of
the
Burnside
ring
of
a
fusion
system
✩Matthew Gelvina, Sune Precht Reehb, Ergün Yalçınc,∗
aMathematicsandComputerScienceDepartment,Wesleyan University,
Middletown,CT06459-0128,USA
b
DepartmentofMathematicalSciences,UniversityofCopenhagen, Copenhagen, Denmark
cDepartmentofMathematics,BilkentUniversity,06800Bilkent,
Ankara,Turkey
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received 23 May 2014
Available online 12 November 2014 Communicated by Michel Broué
Keywords:
Burnside ring Fusion systems Mobius function
WeconsidertheBurnsideringA(F) ofF-stableS-setsfora saturatedfusionsystemF definedonap-groupS.Itisshown byS.P. ReehthatthemonoidofF-stablesetsisafree commu-tativemonoidwithcanonicalbasis{αP}.Wegiveanexplicit
formulathatdescribesαP asanS-set.Intheformulaweuse
acombinatorialconceptcalledbrokenchainswhichwe intro-ducetounderstandinversesofmodifiedMöbiusfunctions.
© 2014ElsevierInc.All rights reserved.
1. Introduction
ForafinitegroupG,theBurnsideringA(G) isdefinedastheGrothendieckringofthe
isomorphismclassesof G-sets with additiongiven by disjoint unionand multiplication
✩ The research is supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). The third author is also partially supported by TÜBİTAK – TBAG/110T712.
* Corresponding author.
E-mailaddresses:mgelvin@wesleyan.edu(M. Gelvin), spr@math.ku.dk(S.P. Reeh), yalcine@fen.bilkent.edu.tr(E. Yalçın).
http://dx.doi.org/10.1016/j.jalgebra.2014.10.026
by Cartesian product. The Burnside ring A(G) isfree as an abelian group, with basis
givenbyisomorphismclassesoftransitiveG-sets[G/H].Inparticularthebasiselements
are inone-to-onecorrespondencewithG-conjugacyclassesofsubgroupsof G.
OneoftenstudiestheBurnsideringofafinitegroupG usingthemarkhomomorphism
Φ : A(G) → ZCl(G), where Cl(G) is the set of G-conjugacy classesof subgroups of G.
For K≤ G, theK’th coordinateof Φ isdefinedbyΦK(X)=|XK| whenX isaG-set,
extendedlinearlyfortherestofA(G).TheringA∗(G):=ZCl(G)istheringofsuperclass
functions f : Cl(G) → Z with multiplication given by coordinate-wise multiplication.
It is called the ghost ring of G and it plays an important role for explaining G-sets
using their fixed point data. In particular, it is shown that the mark homomorphism
is an injectivemap with a finite cokernel. This means that using rational coefficients,
one can express the idempotent basis of A∗(G) in terms of basiselements [G/H] (see
D. Gluck [6]).
Given asaturatedfusionsystemF on ap-groupS, onecandefinetheBurnside ring
A(F) ofthefusionsystem F asasubringof A(S) formedby elements X ∈ A(S) such thatΦP(X) = Φϕ(P )(X) foreverymorphism ϕ: P → S in F. Thissubringis alsothe
GrothendieckringofF-stableS-sets(see(2.3)foradefinition).ItisprovedbyS.P. Reeh
[10] thatthemonoid ofF-stableS-sets is afree commutativemonoidwith acanonical
basis satisfying certain properties. Our primary interest is to identify the elements of
this basis,sowedescribeitinmoredetailhere.
For every X ∈ A(S), let cQ(X) denote the number of [S/Q]-orbits in X so that
X = cQ(X)[S/Q], where the sum is taken over the set of S-conjugacy classes of
subgroups of S.ForeachF-conjugacy classof subgroupsP of S, there isaunique(up
to S-isomorphism)F-stableset αP satisfying
(i) cQ(αP)= 1 ifQ isfullynormalizedandF-conjugateto P ,
(ii) cQ(αP)= 0 ifQ isfullynormalizedandnotF-conjugatetoP .
Theset{αP} overallF-conjugacyclassesofsubgroupsforman(additive)basisforA(F) (see Proposition 2.2).
The main purpose of this paper is to give explicitformulas for the numberof fixed
points|(αP)Q| andforthecoefficientscQ(αP) of[S/Q]-orbits,forthebasiselement αP.
Our first observation is that the matrix of fixed points FMarkQ,P = |(αP)Q| can
be described using a simple algorithm in linear algebra. We now explain this
algo-rithm.
Let Möb = Mark−1 denote theinversematrixof theusualtableof marksfor S.For each F-conjugacy class of subgroups of S, takethe sumof the corresponding columns
of Möb, obtaining a non-square matrix. Then, from the set of rows corresponding to
an F-conjugacy class, select one representing a fully F-normalized subgroup; delete
the others. TheresultingmatrixFMöb is asquarematrix withdimension equalto the
FMark := FMöb−1 is the matrix of marks for A(F). In other words, we prove the following:
Theorem 1.1. Let F be a saturated fusion system over a finite p-group S. Let the
square matrix FMöb be constructed as above, with rows and columns corresponding to the F-conjugacy classes of subgroup in S. Then FMöb is invertible, and the inverse FMark := FMöb−1 is thematrix ofmarks forA(F),i.e.
FMarkQ∗,P∗=(αP∗)Q
∗ .
Here Q∗ and P∗ denotethe chosenF-conjugacy class representatives.This theorem
isproved asTheorem 3.1 inthepaper.We alsogiveadetailedcalculation toillustrate
thismethod(seeExample 3.2).ThisisalldoneinSection3.
InSection4welookcloselyattheabovematrixmethodandanalyzeitusingMöbius
inversion. We observe that the entries of FMark, the table of marks for F, can be
explained by a combinatorial formula using a concept called (tethered) broken chains
(seeDefinition 4.7).ThisformulaisprovedinTheorem 4.9.
InSection5,we provethe maintheorem ofthepaper, whichgivesaformulaforthe
coefficients cQ(αP) in the linearcombination αP =
cQ(αP)[S/Q]. As inthe case of
fixed pointorders, herealso theformula isgiven intermsof analternatingsumof the
numberof brokenchainslinkingQ toP (see Definition 5.1).Themain theorem of the
paperisthefollowing:
Theorem1.2.LetF beasaturatedfusionsystemoverafinitep-groupS.LetBCF(Q,P )
denotethe set of F-broken chains linkingQ to P . Then thenumber of [S/Q]-orbits in each irreducibleF-stable setαP,denoted cQ(αP),can be calculatedas
cQ(αP) = |WS P∗| |WSQ| · σ∈BCF(Q,P ) (−1)(σ)
forQ,P subgroups of S, whereP∗∼FP isfullynormalized.
Intheaboveformula, (σ) denotesthelengthof abrokenchainσ = (σ0,σ1,. . . ,σk)
linkingQ to P defined as theinteger (σ):= k +|σ0|+· · · + |σk| (see Definition 5.1).
Thistheorem isprovedinSection5as Theorem 5.2.InExample 5.5,we illustratehow
this combinatorial formula can be used to calculate the coefficients cQ(αP) for some
subgroupsQ,P forthefusionsystemF = FD8(A6).
InSection6,weprovesomesimplificationsfortheformulainTheorem 1.2.These
sim-plificationscomefrom observationsaboutbroken chainsand frompropertiesofMöbius
functions.TheninSection7wegiveanapplicationofourmaintheoremtocharacteristic
bisets.Sinceunderstandingthecharacteristic bisetswasoneofthemotivationsfor this
LetS beap-groupandF beafusionsystemon S asbefore.A characteristicbisetfor
thefusionsystemF isan(S,S)-bisetΩ satisfyingcertainproperties(seeDefinition 7.1).
ThesebisetswerefirstintroducedbyLinckelmannandWebb,andtheyplayanimportant
role in fusion theory. One of the properties of a characteristic biset is stability under
F-conjugation,namelyforeveryϕ: Q→ S,the(Q,S)-bisetsϕΩ andQΩ areisomorphic.
Since each (S,S)-biset is aleft (S× S)-set, we can convert this stability condition to
a stability condition for the fusion system F × F on the p-group S× S and consider
characteristic bisetsaselementsinA(F × F).
It isshownbyM. Gelvinand S.P. Reeh[5]thateverycharacteristic bisetincludesa
uniqueminimalcharacteristicbiset,denotedbyΩmin.Theminimalbisetcanbedescribed
as the basis element αΔ(S,id) of the fusion system A(F × F), where for a morphism
ϕ : Q → S in F, the subgroup Δ(P,ϕ) denotes the diagonal subgroup {(ϕ(s),s) | s ∈ P } in S× S. Now Theorem 1.2 can be used to give formulas for the coefficients cΔ(P,ϕ)(Ωmin).Suchformulasareimportantforvariousotherapplicationsofthesebisets
(seeforexample[12]).Usingthenewinterpretationofthesecoefficientswewereableto
give aproof for the statementthat allthe stabilizers Δ(P,ϕ) appearing inΩmin must
satisfy P ≥ Op(F) whereOp(F) denotesthelargest normalp-subgroupof F.This was
originallyprovedin[5,Proposition 9.11],theproofwegiveinProposition 7.3usesbroken
chainsandismuchsimpler.
2. Burnsideringsforgroupsandfusionsystems
InthissectionwerecalltheBurnsideringofafinitegroupS andhowto describeits
structureintermsofthehomomorphismofmarks,whichembedstheBurnsideringinto
asuitableghostring.WealsorecalltheBurnsideringofasaturatedfusionsystemF on
ap-groupS,inthesenseof[10].
LetS beafinitegroup.WeusetheletterS insteadof G forafinitegroupsinceinall
theapplicationsoftheseresultsthegroupS willbeap-group.Theisomorphismclasses
of finite S-sets form a semiring with disjoint unionas addition and Cartesian product
as multiplication.TheBurnsideringof S,denotedA(S),isthendefinedastheadditive
Grothendieckgroupofthissemiring,andA(S) inheritsthemultiplicationaswell.Given
a finiteS-set X,welet [X] denotethe isomorphismclass ofX as an element ofA(S).
Theisomorphismclasses[S/P ] oftransitiveS-sets formanadditivebasisforA(S),and
two transitive setsS/P and S/Q are isomorphicifand only ifthesubgroups P and Q
are conjugatein S.
ForeachelementX ∈ A(S) wedefinecP(X),withP ≤ S,tobethecoefficientswhen
we writeX asalinearcombination ofthebasiselements [S/P ] inA(S),i.e.
X =
[P ]∈Cl(S)
cP(X)· [S/P ],
whereCl(S) denotesthesetofS-conjugacyclassesofsubgroupinS.Theresultingmaps
To describe the multiplication of A(S), it is enough to know the products of basis
elements[S/P ] and[S/Q].BytakingtheCartesian product(S/P )× (S/Q) and
consid-eringhow it breaks into orbits,one reaches thefollowing double cosetformula for the
multiplicationinA(S):
[S/P ]· [S/Q] =
s∈P \S/Q
S/(P ∩sQ), (2.1)
whereP\S/Q isthesetofdouble cosetsP sQ withs∈ S.
Insteadofcountingorbits,analternativewayofcharacterizing afiniteS-setis
count-ingthefixedpointsforeachsubgroupP ≤ S.ForeveryP ≤ S andS-set X,wedenote
thenumberofP -fixedpointsbyΦP(X):=|XP|.Thisnumberonlydependson P upto
S-conjugation.Since wehave
(X Y )P=XP+YP and (X× Y )P=XP · YP
forallS-setsX andY ,thefixedpointmap ΦP forS-setsextendstoaringhomomorphism
ΦP : A(S)→ Z. Onthebasiselements [S/P ],thenumberoffixed pointsisgivenby
ΦQ
[S/P ]=(S/P )Q= |NS(Q, P )|
|P | , (2.2)
whereNS(Q,P )={s∈ S |sQ≤ P } isthetransporterinS fromQ toP .Inparticular,
ΦQ([S/P ]) = 0 ifandonlyifQS P (Q is subconjugatetoP ).
Wehave onefixed point homomorphism ΦP per conjugacy class ofsubgroups in S,
andwecombinetheminto thehomomorphism ofmarks
Φ = ΦS: A(S) [P ]ΦP −−−−−→ [P ]∈Cl(S) Z.
This ringhomomorphismmaps A(S) into theproduct ringA∗(S):= [P ]∈Cl(S)Z, the
so-calledghostring for theBurnside ringA(S).
WethinkoftheelementsintheghostringA∗(S) as superclassfunctionsCl(S)→ Z
definedonthesubgroupsof S andconstantoneveryS-conjugacyclass. Foranelement
ξ∈ A∗(S) wewriteξ(Q),withQ≤ S,todenotethevalueoftheclassfunctionξ onthe
S-conjugacy class of Q. We thinkof ξ(Q) as the numberof Q-fixed points for ξ, even
thoughξ might notbe thefixedpoint vectorforan actualelement ofA(S). Theghost
ring A∗(S) has a naturalbasis consisting of eP for each [P ] ∈ Cl(S), where eP is the
classfunctionwithvalue1 ontheclass[P ],and0 onalltheotherclasses.Theelements
{eP | [P ]∈ Cl(S)} aretheprimitiveidempotentsof A∗(S).
ResultsbytomDieckandothersshowthatthemarkhomomorphismisinjective,but
containstheobstruction toξ beingthefixedpointvector ofa(virtual) S-set,hencewe
speakofthiscokernelastheobstructiongroup Obs(S):= [P ]∈Cl(S)(Z/|WSP|Z),where
WSP := NSP/P .Thesestatementsarecombinedinthefollowingproposition,theproof
of whichcanbefound in[2,Chapter 1],[3],and[13].
Proposition 2.1. LetΨ = ΨS : A∗(S)→ Obs(S) be givenby the[P ]-coordinatefunctions
ΨP(ξ) := s∈WSP ξs P mod |WSP| .
Then, thefollowingsequenceof abeliangroups isexact: 0→ A(S)−→ AΦ ∗(S)−→ Obs(S) → 0.Ψ
Note that in the exact sequence above Φ is a ring homomorphism, but Ψ is just a
grouphomomorphism.
2.1. TheBurnsidering of asaturatedfusion system
LetS beafinitep-group,andsupposethatF isasaturatedfusionsystemon S (see[1]
fornecessary definitionsonfusionsystems).WesaythatafiniteS-set isF-stableifthe
actionisunchangeduptoisomorphismwheneverweactthroughmorphismsofF.More
precisely, if P ≤ S is a subgroup and ϕ : P → S is a homomorphism in F, we can
considerX asaP -setbyusingϕ todefinetheactiong.x:= ϕ(g)x forg∈ P .Wedenote
the resulting P -set by P,ϕX. In particular when incl : P → S is the inclusion map,
P,inclX hastheusual restrictionoftheS-actionto P .
Restrictingtheactionof S-setsalongϕ extendstoaringhomomorphismrϕ: A(S)→
A(P ),andweletP,ϕX denotetheimagerϕ(X) forallelementsX∈ A(S). Wesaythat
anelement X ∈ A(S) isF-stable ifitsatisfies
P,ϕX =P,inclX inside A(P ), for all P ≤ S and
homomorphisms ϕ : P → S in F. (2.3)
The F-stability condition originally came from considering action maps S → ΣX into
thesymmetricgrouponX thataremapsoffusionsystemsF → FΣX.
Alternatively,onecancharacterizeF-stabilityintermsof fixedpointsand themark
homomorphism, andthefollowingthreepropertiesareequivalentforallX ∈ A(S):
(i) X isF-stable.
(ii) ΦP(X)= ΦϕP(X) for allϕ∈ F(P,S) andP ≤ S.
A proof ofthis claim canbe found in[4, Proposition 3.2.3] or [10]. Weshall primarily
use(ii)and(iii)tocharacterizeF-stability.
Itfollows fromproperty (iii)thattheF-stableelementsform a subringof A(S).We
define the Burnside ring of F to be the subring A(F) ⊆ A(S) consisting of all the
F-stable elements. Equivalently, we can consider the actual S-sets that are F-stable: TheF-stable sets form a semiring, and we define A(F) to be the Grothendieckgroup
hereof.ThesetwoconstructionsgiverisetothesameringA(F) –see[10].
Accordingto [10], everyF-stableS-set decomposesuniquely (upto S-isomorphism)
as adisjoint union ofirreducible F-stable sets, where theirreducible F-stable sets are
thosethatcannotbewrittenasdisjointunionsofsmallerF-stablesets.Eachirreducible
F-stablesetcorrespondstoanF-conjugacyclass[P ]F ={Q≤ S | Q is isomorphic to P inF} ofsubgroups,and theysatisfythefollowingcharacterization:
Proposition2.2.(See[10, Proposition 4.8 and Theorem A].) LetF beasaturatedfusion
systemoverS.ForeachconjugacyclassinF ofsubgroups [P ]F thereisaunique(upto S-isomorphism)F-stableset αP satisfying
(i) cQ(αP)= 1 ifQ is fullynormalizedandF-conjugatetoP ,
(ii) cQ(αP)= 0 ifQ is fullynormalizedandnotF-conjugate toP .
ThesetsαP formanadditivebasisforthemonoidofallF-stableS-sets.Inaddition,by constructionin [10]thestabilizerof anypointinαP isF-conjugatetoasubgroupofP .
3. FixedpointordersoftheirreducibleF-stablesets
LetMark be the matrixof marksfor theBurnside ringof S, i.e.the matrixfor the
markhomomorphismΦ: A(S)→ A∗(S) withentries
MarkQ,P =(S/P )Q= |N
S(Q, P )| |P | .
Therowsand columns of Mark correspond to theS-conjugacy classes[P ]S ∈ Cl(S) of
subgroups in S. We order the subgroup classes by increasing order of the subgroups,
inparticular the trivial group1 corresponds to the first row and column,and S itself
correspondsto thelast rowandcolumn.This wayMark becomesuppertriangular.
OvertherationalnumbersthemarkhomomorphismΦ: A(S)⊗ Q−→ A∼= ∗(S)⊗ Q is
anisomorphism,andweletMöb = Mark−1 be theinverserationalmatrix.
FromMöb weconstructafurthermatrixFMöb asfollows:ForeachF-conjugacyclass
of subgroups in S we take the sum of the corresponding columns of Möb to be the
columnsofFMöb.ForeachF-conjugacyclassofsubgroupswechooseafullynormalized
representative of the class, and then we delete allrows thatdo not correspond to one
matrix with dimension equal to the number of F-conjugacy classes of subgroups; the
rows and columns correspond to the chosenrepresentatives ofthe F-conjugacy classes
(see Example 3.2).
Foreachclass[P ]F letP∗ bethechosenrepresentative.Theprecisedescriptionofthe
entries FMöbQ∗,P∗ intermsofMöbQ,P isthen
FMöbQ∗,P∗:=
[P ]S⊆[P∗]F
MöbQ∗,P.
Theorem 3.1. Let F be a saturated fusion system over a finite p-group S. Let the
square matrix FMöb be constructed as above, with rows and columns corresponding to the F-conjugacy classes of subgroup in S. Then FMöb is invertible, and the inverse FMark := FMöb−1 isthematrixof marks forA(F),i.e.
FMarkQ∗,P∗=(αP∗)Q∗.
Proof. IntherationalghostringA∗(S)⊗Q= [P ]SQ theunitvectoreP isthesuperclass
functionwithvalue1 fortheclass[P ] andvalue0 fortheothersubgroupclasses.Wehave
oneunitvectoreP corresponding toeachconjugacyclass[P ]S,andthematrixofmarks
Mark expressesthe usualbasisfor A(S)⊗ Q,consisting of thetransitivesets[S/P ], in
termsoftheidempotentseP.Conversely,theinverseMöb = Mark−1 thenexpressesthe
idempotents eP as(rational)linearcombinations oftheorbits[S/P ].
AnelementX∈ A(S)⊗Q isF-stableifthenumberoffixedpoints|XQ| and|XP| are
the samefor F-conjugate subgroups Q∼F P ,i.e. if thecoefficients of X with respect
to the idempotents eQ and eP are the same forF-conjugate subgroups. The F-stable
elements ofA∗(F)⊗ Q≤ A∗(S)⊗ Q thushaveanidempotent basisconsisting of
eFP :=
[P]S⊆[P ]F eP,
with a primitive F-stable idempotent eFP corresponding to each F-conjugacy class of
subgroups. To express the idempotent eFP as alinear combination of orbits [S/P ], we
justhavetotakethesumofthecolumnsinMöb associatedtotheconjugacyclass[P ]F.
Hencecountingthenumberof[S/Q]-orbitsineFP, weget
cQ
eFP=
[P]S⊆[P ]F
MöbQ,P.
LetP∗denotethesetofchosenfullynormalizedrepresentativesforeachF-conjugacy
classofsubgroupsinS.ForeachP∗∈ P∗,wehaveanirreducibleF-stablesetαP∗,and
byProposition 2.2 anylinearcombination X ofthe{αP∗} canbe determinedsolely by
counting thenumberof[S/P∗]-orbitsforeachP∗∈ P∗.
Because eFP is F-stable, it is a (rational)linear combination of the {αQ∗}. The
expressthe idempotentseFP intermsof the{αQ∗} weonly careabouttherowsofMöb
corresponding to Q∗ ∈ P∗, and ignore all the other rows. Consequently, FMöb is the
matrixthatexpressestheidempotentseFP intermsofthe{αQ∗}.
The inverseFMark = FMöb−1 therefore expresses the irreducible F-stable sets αP
intermsoftheF-stableidempotentseQF,whichexactlyreduces tocountingtheQ-fixed
pointsofαP. 2
Example3.2.LetS = D8 bethedihedralgroupoforder 8andF = FD8(A6) denotethe
fusionsysteminducedbythefinitegroupA6.LetP betheentiresubgroupposetofD8
andP/F theposetofF-conjugacyclassesofsubgroupsinD8:
P P/F D8 [D8] V1 4 C4 V42 [V41] [C4] [V42] C21 C21 F Z C 2 2 F C 2 2 [Z] 1 [1]
where V4∗ is anelementary abelian2-group oforder 4, Cn∗ isacyclic groupof order n,
Z ∼= C2 is the centre of D8, and square brackets denote the F-conjugacy class. The
horizontalsquigglylines indicatesubgroups’ being inthe sameD8-conjugacyclass and
dashedlinesmeansthattheyareinthesameF-conjugacy class.
ThetableofmarksMark anditsinverseMöb = Mark−1 aregivenbelow
Mark 1 C21 Z C22 V41 C4 V42 D8 1 8 4 4 4 2 2 2 1 C21 2 0 0 2 0 0 1 Z 4 0 2 2 2 1 C2 2 2 0 0 2 1 V1 4 2 0 0 1 C4 2 0 1 V42 2 1 D8 1
Möb 1 C1 2 Z C22 V41 C4 V42 D8 1 1/8−1/4 −1/8 −1/4 1/4 0 1/4 0 C1 2 1/2 0 0 −1/2 0 0 0 Z 1/4 0 −1/4 −1/4 −1/4 1/2 C22 1/2 0 0 −1/2 0 V1 4 1/2 0 0−1/2 C4 1/2 0−1/2 V2 4 1/2−1/2 D8 1
Below we givethe matrixfor FMöb andits inverseFMark = FMöb−1. Recallthat the
matrixforFMöb isobtainedbyaddingthecolumnsofMöb forthesubgroupswhichare
F-conjugate, and then by choosing a fully normalized subgroup in everyF-conjugacy
class on the rows. Here Z is theunique fully normalizedsubgroup inits F-conjugacy
class. FMöb 1 Z V41 C4 V42 D8 1 1/8−5/8 1/4 0 1/4 0 Z 1/4−1/4 −1/4 −1/4 1/2 V1 4 1/2 0 0 −1/2 C4 1/2 0 −1/2 V2 4 1/2 −1/2 D8 1 FMark 1 Z V1 4 C4 V42 D8 1 8 20 6 10 6 1 Z 4 2 2 2 1 V41 2 0 0 1 C4 2 0 1 V2 4 2 1 D8 1
From this we obtain the Φ(αP) by reading off the columns of FMark (since eFZ =
eC1 2 + eZ+ eC22): Φ(α1) = 8eF1 = 8e1 Φ(αZ) = 20eF1 + 4eFZ = 20e1+ 4eC1 2+ 4eZ+ 4eC22 Φ(αV1 4) = 6e F 1 + 2eFZ + 2eFV1 4 = 6e1+ 2eC 1 2 + 2eZ+ 2eC22+ 2eV41 Φ(αC4) = 10eF1 + 2eFZ + 2eFC4 = 10e1+ 2eC21+ 2eZ+ 2eC22+ eC4 Φ(αV2 4) = 6e F 1 + 2eFZ + 2eFV2 4 = 6e1+ 2eC 1 2 + 2eZ+ 2eC22+ 2eV42 Φ(αD8) = eF1 + eFZ + eFV1 4 = e1+ eC 1 2 + eZ+ eC22 + eFC4+ eFV2 4 + e F D8 + eV41+ eC4+ eV42+ eD8
Finally,applyingthematrixMöb toeachofthese fixedpoint vectorsyieldstheS-orbit descriptionoftheαP: α1= [S/1] αZ = [S/Z] + 2 S/C21+ 2S/C22 αV1 4 = S/V41+S/C22 αC4 = [S/C4] + S/C21+S/C22 αV2 4 = S/V42+S/C21 αD8 = [S/D8]
Thereisanexplicitformulaforexpressingtheidempotentbasis{eQ} intermsofthe
transitiveS-set basis{[S/P ]} using the combinatoricsof thesubgroup poset, which is
often referred as the Gluck’s idempotent formula [6]. In the following two sections we
findsimilar explicitformulasfor thecoefficientsofαP∗ withrespect tothe idempotent
basis{eQ} andthenwithrespecttotheS-setbasis{[S/P ]}.Forthisweneedtolookat
theMöbiusinversioninGluck’sidempotentformulamoreclosely.
4. FixedpointordersandMöbiusinversion
Inthissectionwediscusshowamoreexplicitformulacanbeobtainedforfixedpoint
ordersofbasiselementsusingMöbiusinversion.Wefirstintroducebasicdefinitionsabout
Möbiusinversion.Formoredetails,wereferthereaderto[11].
LetP beafinite poset.Theincidencefunctionof P isdefinedas thefunction
ζP :P × P → Z : (a, b) →
1 a≤ b,
0 else.
Theincidence matrix ofP isthe |P|× |P|-matrix (ζP) with entries(ζP)a,b = ζP(a,b).
Whenlabelling therows/columnswerespect thepartial orderof P,suchthata≤ b in
P impliesthatthe a-row/-columnprecedes theb-row/-column. This waythe incidence
matrixisalwaysupperunitriangular(anuppertriangularmatrixwithalldiagonalentries
equalto 1).
Definition4.1.TheMöbiusfunction for aposetP isμP :P × P → Q defined by
a∈P
ζP(x, a)μP(a, y) = δx,y =
a∈P
μP(x, a)ζP(a, y)
forallx,y∈ P.If thecorresponding |P|× |P| Möbius matrix is(μP),wehave(μP)=
(ζP)−1.
Proof. Byourlabellingconvention,(ζP) isupperunitriangular.Thereforewecanwrite
(ζP) = I + (ηP), where (ηP)i,j = 1 when ai < aj and vanishes elsewhere. Then ηP is
strictly uppertriangular,and(ηP)|P|= 0,so
(μP) = (ζP)−1=I + (ηP)−1
= I− (ηP) + (ηP)2− (ηP)3+ . . . + (−1)|P|−1(ηP)|P|−1
hasallintegralentries. 2
Each of the matrices (ηPk) := (ηP)k hasan interpretation interms of chains in the
posetP.
Definition 4.3. A chain of length k in P is atotally ordered subset of k + 1 elements
σ ={a0< a1< . . . < ak}.Suchachainlinks a0 toak.
LetCkP(a,b) bethesetofchainsoflengthk linkinga tob,andCkP thesetofallchains
of lengthk inP. C0P is theset ofelements ofP.Similarly,letCP(a,b) betheset of all
chains linking a to b, CP the set of all chains in P, and for any chain σ ∈ CP let |σ|
denote thelengthof σ.
Lemma4.4.(ηkP)a,b=|CkP(a,b)|.
Proof. With a0:= a and ak := b,thedefinitionofmatrixmultiplicationgivesus
ηPka 0,ak= a1,...,ak−1∈P ηP(a0, a1)· ηP(a1, a2)· · · ηP(ak−1, ak).
BydefinitionoftheincidencefunctionηP,eachfactorηP(ai,ai+1) is1 ifai< ai+1 and
zero otherwise. The product ηP(a0,a1)· · · ηP(ak−1,ak) is therefore nonzero and equal
to 1 precisely whena0< a1<· · · < ak isak-chaininP linkinga0 toak. 2
Proposition 4.5. Foralla,b∈ P,
μP(a, b) = ∞ k=0 (−1)kCkP(a, b)= σ∈CP(a,b) (−1)|σ|.
Proof. Immediatefrom Lemmas 4.2 and4.4andtheirproofs. 2
Remark4.6.FromtheformulaaboveitisclearthattheMöbiusfunctioncanbeexpressed
as the reducedEuler characteristic of asubposet inP.More specifically,for a< b, let
(a,b)P denote the poset of all c ∈ P with a < c < b. Then μP(a,b) is equal to the
reduced Euler characteristic χ((a,b)P) of the subposet (a,b)P for every a,b ∈ P such
4.1. Möbiusfunctionsandfixed points
Onthenextfewpageswe gothroughtheconstructionofthematrixFMark in
The-orem 3.1 again, butthis time we follow the calculationsin detailusing the framework
ofincidenceandMöbiusfunctions.Forafinite p-groupS,weletP betheposetof
sub-groupsordered byinclusion.Thisposethasincidence andMöbiusfunctionsζP and μP
asdescribed intheprevioussection,denotedthereafterbyζ,respectivelyμ.
The matrix of marks Mark for the Burnside ring of S has entries Mark[Q],[P ] =
|NS(Q,P )|/|P | defined for pairs ([Q],[P ]) of S-conjugacy classes of subgroups in S.
Each column is divisible by the diagonal entry, which is the order of the Weyl group
WSP = NSP/P .Ifwedividethe[P ]S-columnby|WSP|,weget
Mark[Q],[P ]· 1 |WSP| = |NS(Q, P )| |NSP| = |{s ∈ S | sQ≤ P }| |NSP| =|{s ∈ S | Q ≤ P s}| |NSP| = P≤ SP∼S P and Q≤ P= P∼SP ζQ, P.
We denote this value by ζS([Q],[P ]), and we call ζS the modified incidence function
for theS-conjugacy classesof subgroups. We have( ζS)[Q],[P ] = Mark[Q],[P ]/|WSP|, so
the modified incidence matrix ( ζS) is upper unitriangular (see Example 4.10 for the
computationsinthecaseS = D8).
Inverting the matrix ( ζS), we define ( μS) := ( ζS)−1 which gives rise to a modified
Möbius function μS forS-conjugacy classes of subgroups. Since Möb = Mark−1 is the
inverseofthematrixofmarks,wehave( μS)[Q],[P ]=|WSQ|·MöbQ,P.As( ζS) istriangular
withdiagonalentries1,wealsohave( μS)= ( ζS)−1=
∞
k=0(−1)k· (( ζS)− I)k asinthe
proofofLemma 4.2,whichweusetocalculatetheentriesof( μS):
μS[Q], [P ]= ∞ k=0 (−1)k·( ζS)− I k = ([R0],[R1],...,[Rk])∈TS (−1)k ζS [R0], [R1] · · · ζS [Rk−1], [Rk]
whereTS consistsofalltuples ([R0],[R1],. . . ,[Rk]), fork≥ 0,ofS-conjugacyclassesof
subgroups[Ri]∈ Cl(S) suchthat[R0]= [Q],[Rk]= [P ],and|R0|<|R1|<· · · < |Rk|.
Sincewehave ζS [Ri], [Rj]= Rj∼SRj ζRi, Rj
([R0],[R1],...,[Rk])∈TS R0=Q R1∼SR1 (−1)kζR0, R1 ζS R1, [R2] · · · ζS [Rk−1], [Rk] .. . = ([R0],[R1],...,[Rk])∈TS R0=Q R1∼SR1 R2∼SR2 · · · × Rk∼SRk (−1)kζR0, R1 ζR1, R2· · · ζRk−1, Rk = R0<R1<···<Rk s.t. R0=Q, Rk∼SP (−1)k = P∼SP ∞ k=0 (−1)kCkPQ, P= P∼SP μQ, P.
Therefore, thematrixMöb,theinverseofthematrixofmarks,hasentries
Möb[Q],[P ]= 1 |WSQ| μS [Q], [P ]= 1 |WSQ| P∼SP μQ, P.
Thisconcludesthepartofourinvestigationconcerningonlythesubgroupstructureof S,
andforthecalculationsbelowweincludetheextradataofasaturatedfusionsystemF
on S.
In order to determine the number of fixed points |(αP)Q| as in Theorem 3.1, we
wish to calculate theF-analogs ofMark and Möb above. To dothis, we first choosea
fullynormalizedrepresentativeP∗foreachF-conjugacyclass[P ]F ofsubgroups,andas
before letP∗ be thecollectionofthese representatives.RecallthatthematrixFMöb is
constructedfromMöb bypickingouttherowscorrespondingtoQ∗∈ P∗,andthecolumn
inFMöb correspondingto P∗ ∈ P∗ isthesumof thecolumnsinMöb correspondingto
[P ]S withP ∼F P∗.Moreexplicitly,wehave
FMöbQ∗,P∗ := [P ]S⊆[P∗]F Möb[Q∗],[P ]= 1 |WSQ∗| P∼FP∗ μQ∗, P.
We definethe modifiedMöbius function μF :P∗× P∗ → Z forthe(representatives of)
F-conjugacy classesofsubgroups,tobe
μFQ∗, P∗:=WSQ∗ · FMöbQ∗,P∗ =
P∼FP∗
μQ∗, P,
summing the usual Möbiusfunction. The associated matrix( μF) isthen upper
unitri-angular.
The modified incidence matrixfor F is definedas the inverse( ζF):= ( μF)−1, with
theassociated function ζF :P∗× P∗→ Z.ByTheorem 3.1wethenhave
(αP)Q=FMark
where FMark := FMöb−1.Recall thatfor eachsubgroup R ≤ S, wedenote by R∗ the
chosenfullynormalizedrepresentativefortheF-conjugacyclassofR.Aspreviously,the
factthat( μF) isunitriangularimplies
ζFQ∗, P∗= ∞ k=0 (−1)k·( μF)− Ik = (R∗0,R∗1,...,Rk∗)∈TF (−1)k μFR0∗, R1∗ · · · μFR∗k−1, R∗k
where TF consists of all tuples (R∗0,R1∗,. . . ,R∗k), for all k ≥ 0, of F-conjugacy class
representatives R∗i ∈ P∗ suchthatR∗0 = Q∗, Rk∗ = P∗, and |R0∗| <|R∗1|<· · · < |R∗k|.
Sincewehave μFR∗i, R∗j= Rj∼FR∗j μR∗i, Rj ,
forallR∗i,R∗j ∈ P∗,weobtainthat
ζF(Q∗, P∗) = (R∗0,R1∗,...,R∗k)∈TF R1∼FR∗1 (−1)kμR∗0, R1 μFR∗1, R∗2 · · · μFR∗k−1, R∗k .. . = (R∗0,R∗1,...,R∗k)∈TF, R1,...,Rk∈P s.t. Ri∼FR∗i (−1)kμR∗0, R1 μR∗1, R2 · · · μR∗k−1, Rk = (R∗0,R∗1,...,R∗k)∈TF, R1,...,Rk∈P s.t. Ri∼FR∗i σi∈CP(R∗i−1,Ri) for 1≤i≤k (−1)k+|σ1|+···+|σk|. (4.1)
Tocalculate ζF(Q∗,P∗) wehencehavetocountsequencesofchains(σ1,. . . ,σk) such
thattheendRi ofσi isF-conjugateto thestartR∗i ofσi+1,and thefirstchainσ1 has
to startat Q∗ whilethefinal chainσk only hasto endat P∗ upto F-conjugation.We
givethesesequencesaname:
Definition4.7.AtetheredF-brokenchaininP linkingQ∗∈ P∗toP ∈ P isasequenceof
chains(σ1,. . . ,σk) inP subject tothefollowingrequirements.Witheachchainwritten
asσi= (ai0,. . . ,aini) theymustsatisfy
• ai
ni ∼F a
i+1
0 for all1≤ i≤ k − 1, sothe endpointsof thechainsfit togetherupto
conjugation inF.
• ai
0∈ P∗ forall1≤ i≤ k. Everychainstartsat oneofthechosenrepresentatives.
• |σi|= ni> 0,forall1≤ i≤ k.
If Q∗∼F P ,we allowthetrivialbrokenchainwith k = 0.LettBCF(Q∗,P ) be theset
oftetheredF-brokenchainslinkingQ∗toP .Thetotallength ofatetheredbrokenchain
σ = (σ1,. . . ,σk) isdefinedtobe
(σ) := k +|σ1| + · · · + |σk|.
Wevisualize atethered brokenchainas azigzagdiagram inthefollowingway:
a10 · · · a1n1 a2 0 · · · a2n2 . .. ak 0 · · · aknk < < ∼ < < ∼ ∼ < <
Thetotallengthofthetetheredbrokenchainisthenthetotalnumberof< and∼ signs
plus 1. The added 1 can be viewed as an additional hidden Q ∼F Q∗ infront of the
broken chain,and this interpretationmatchesthe description,inRemark 5.3below,of
tethered brokenchainsas aspecialcaseofthebroken chains definedinSection5.
With theterminology of tethered brokenchains, thecalculations abovetranslateto
thefollowing statements:
Proposition 4.8. The modified incidence function ζF for a saturated fusion system F,
can be calculatedas ζFQ∗, P∗= σ∈tBCF(Q∗,P∗) (−1)(σ)= (σ1,...,σk)∈tBCF(Q∗,P∗) (−1)k+|σ1|+···+|σk|
forallfullynormalized representativesQ∗,P∗∈ P∗.
Wenow statethemain resultof thissection.
Theorem4.9.LetF beasaturatedfusionsystemoverafinitep-groupS,andletP∗bea
setoffullynormalizedrepresentativesfortheF-conjugacyclassesofsubgroupsinS.Let
tBCF(Q∗,P∗) denotethesetofalltetheredF-brokenchainslinkingQ∗ toP∗.Thenthe
numbers offixed pointsfortheirreducible F-stablesetsαP∗,P∗∈ P∗,canbe calculated as (αP∗)Q∗=WSP∗ · σ∈tBCF(Q∗,P∗) (−1)(σ) forQ∗,P∗∈ P∗.
Proof. Immediatefromthepropositionsince|(αP∗)Q∗|=|WSP∗|· ζF(Q∗,P∗). 2
Example4.10.LetS = D8andF = FS(A6) asbefore.TheincidencematrixζP andthe
MöbiusmatrixμP aregivenasfollows.
ζP 1 C1 2 C21 Z C2 2 C2 2 V41 C4 V42 D8 1 1 1 1 1 1 1 1 1 1 1 C1 2 1 1 1 C21 1 1 1 Z 1 1 1 1 1 C2 2 1 1 1 C2 2 1 1 1 V41 1 1 C4 1 1 V2 4 1 1 D8 1 μP 1 C1 2 C21 Z C2 2 C2 2 V41 C4 V42 D8 1 1−1 −1 −1 −1 −1 2 0 2 0 C12 1 −1 0 C1 2 1 −1 0 Z 1 −1 −1 −1 2 C2 2 1 −1 0 C22 1 −1 0 V14 1 −1 C4 1 −1 V24 1−1 D8 1
Below we see the matrices for μS and ζS obtained by summing over the columns of
subgroups belongingto the sameS-conjugacyclass and choosing anS-conjugacy class
representativeontherows.
μS 1 C1 2 Z C22 V41 C4 V42 D8 1 1 −2 −1 −2 2 0 2 0 C1 2 1 0 0−1 0 0 0 Z 1 0 −1 −1 −1 2 C22 1 0 0−1 0 V1 4 1 0 0−1 C4 1 0−1 V2 4 1−1 D8 1 ζS 1 C21 Z C22 V41 C4 V42 D8 1 1 2 1 2 1 1 1 1 C21 1 0 0 1 0 0 1 Z 1 0 1 1 1 1 C2 2 1 0 0 1 1 V1 4 1 0 0 1 C4 1 0 1 V42 1 1 D8 1
WD8 1 C 1 2 Z C22 V41 C4 V42 D8 1 8 C1 2 2 Z 4 C22 2 V1 4 2 C4 2 V2 4 2 D8 1 mD8 1 C 1 2 Z C22 V41 C4 V42 D8 1 8 4 4 4 2 2 2 1 C1 2 2 0 0 2 0 0 1 Z 4 0 2 2 2 1 C22 2 0 0 2 1 V1 4 2 0 0 1 C4 2 0 1 V2 4 2 1 D8 1
The lasttwo matrices aboveare thediagonalmatrixWD8 withentries (WD8)[P ],[P ]=
|WS(P )|, and the matrix mD8 = ζS · WD8 which is the same as matrix of the mark
homomorphismMark.SowealsohaveMöb = W−1D8· μS.
Nowlet μF bethematrixobtainedbysummingcolumnsof μS overtheF conjugacy
classesandpickingfullynormalizedrepresentativesfortherows.Let ζF = ( μF)−1.
μF 1 Z V1 4 C4 V42 D8 1 1−5 2 0 2 0 Z 1−1 −1 −1 2 V1 4 1 0 0 −1 C4 1 0 −1 V2 4 1 −1 D8 1 ζF 1 Z V1 4 C4 V42 D8 1 1 5 3 5 3 1 Z 1 1 1 1 1 V1 4 1 0 0 1 C4 1 0 1 V2 4 1 1 D8 1
From the definition of FMöb, it is easy to see thatFMöb = W−1F · μF and FMark =
FMöb−1 = ζ
F · WF where WF is the diagonal matrix with entries (WF)P∗,P∗ =
|WS(P∗)| for all P∗ ∈ P∗. Theorem 4.9 says thatwe can calculate the entries of the
matrix ζF bycountingthenumberoftetheredbrokenchains.Forexample, ζF(1,Z)= 5
because there are 5 tethered broken chainslinking 1 to Z. We givemore complicated
examplesoftetheredbroken chaincalculationsinExample 5.5.
Remark 4.11. Note thatthe modified incidence matrix with respect to S-conjugations
and themodifiedMöbiusfunctiononS-conjugatesubgroups(comingfromtheposetof
subgroups)areconstructedinthesameway:AddthecolumnsofS-conjugatesubgroups,
pick outany row from eachclass. It is interesting thatperforming thesameoperation
on the originals of the incidence function and the Möbius inverse ends up giving you
inverse matrices; inparticular, thisis notwhathappensfor modifications withrespect
to F-conjugation which is what is done in the rest of the paper. We think that this
shows that the S-conjugation action on the subgroup poset is more special that the
5. Brokenchains andthemaintheorem
Now thatwehaveformulas for thenumberof fixed pointsof αP, we will determine
how each αP decomposes into S-orbits. For every element X ∈ A(S) of the Burnside
ring,weletcQ(X) denotethenumberof(virtual)[S/Q]-orbits,i.e.thecoefficientsofthe
linear combination X = [Q]ScQ(X)· [S/Q]. The matrixof marks Mark encodes the
numberoffixedpointsintermsofthenumberoforbits,sothenumbers|XQ| formafixed
pointvectorϕ:= Mark·(cQ(X)).RecallthatMöb istheinverseofMark.Givenanyfixed
pointvectorϕ,wecanthereforerecovertheorbitdecompositionas (cQ(X))= Möb·ϕ.
ForαP we alreadyhaveaformula forthenumberof fixed points|(αP)Q|, which we
writeintheformof
(αP)Q=FMarkQ∗,P∗=WSP∗ · ζFQ∗, P∗
where ζF(Q∗,P∗) has acomplicated Möbiusformula givenin(4.1).We alsoknow how
Möb is given in terms of Möbius functions. The number of [S/Q]-orbits in αP must
thereforebe cQ(αP) = [R]∈Cl(S) MöbQ,R·(αP)R= 1 |WSQ| [R]∈Cl(S) μS[Q], [R]·(αP∗)R∗ = 1 |WSQ| R∈P μ(Q, R)·WSP∗ · ζF R∗, P∗ = |WSP ∗| |WSQ| R∈P μ(Q, R) · (R∗0,R∗1,...,R∗k)∈TF, R1,...,Rk∈P s.t. Ri∼FR∗i (−1)kμR∗0, R1 μR∗1, R2 · · · μR∗k−1, Rk
where thesum isover TF of allk-tuples,for allk ≥ 0, of(prefixed) F-conjugacy class
representatives R∗i ∈ P∗ suchthat R∗0 = R∗, R∗k = P∗, and |R∗0| <|R∗1|<· · · < |R∗k|.
Fromthis weobtainthat
cQ(αP) =|WSP ∗| |WSQ| R0,R1,...,Rk∈P s.t. Rk∼FP∗, |Q|≤|R0|<|R1|<···<|Rk| (−1)kμ(Q, R0)μ R∗0, R1 μR∗1, R2 · · · μRk∗−1, Rk =|WSP ∗| |WSQ| R0,R1,...,Rk∈P s.t. Rk∼FP∗, |Q|≤|R0|<|R1|<···<|Rk| σ0∈CP(Q,R0) σi∈CP(Ri−1∗ ,Ri) for 1≤i≤k (−1)k+|σ0|+|σ1|+···+|σk|.
Theresultingformulaisverysimilartothecalculationsforfixedpointsintheprevious
section,exceptthatwehaveanadditional(possiblytrivial)chainσ0infront.Wecombine
this additional chain with the definition of tethered broken chains and arrive at the
following definition:
Definition 5.1.AnF-brokenchain inP linkingQ∈ P toP ∈ P isasequence ofchains
(σ0,σ1,. . . ,σk) inP subject tothe following requirements.Witheachchainwrittenas σi= (ai0,. . . ,aini) theymustsatisfy
• aini ∼F ai+10 forall 0≤ i≤ k − 1, so theendpoints of thechainsfittogether upto
conjugationinF.
• ai
0 ∈ P∗ for all 1≤ i ≤ k. Every chain except for σ0 starts at one of the chosen
representatives.
• |σi|= ni> 0,forall1≤ i≤ k.Note thatσ0is allowedtobe trivial.
• a0
0= Q andaknk∼F P .
As before, ifQ∼F P ,we allowthetrivial brokenchainwith k = 0 andσ0 trivial.Let
BCF(Q,P ) betheset ofF-brokenchainslinkingQ toP .Wedefine thetotal length of
abrokenchainσ = (σ0,. . . ,σk) tobe
(σ) := k +|σ0| + · · · + |σk|.
To visualizeabrokenchain,werepresentitbythediagram
a0 0 · · · a0n0 a1 0 · · · a1n1 . .. ak 0 · · · aknk < < ∼ < < ∼ ∼ < <
The totallengthoftherepresentedbroken chainisthen equalto thenumberof< and
∼ signsputtogether.
Now westateourmain theorem:
Theorem5.2.LetF beasaturatedfusionsystemoverafinitep-groupS.LetBCF(Q,P )
denote the set of F-broken chains linking Q to P . Thenthe number of [S/Q]-orbits in each irreducibleF-stable setαP,denoted cQ(αP), can becalculated as
cQ(αP) = |WS P∗| |WSQ| · σ∈BCF(Q,P ) (−1)(σ)
forQ,P ∈ P, whereP∗∼FP is fullynormalized.
Proof. Immediatefromtheargumentat thebeginningofthesection. 2
Remark5.3.Ifabrokenchain(σ0,σ1,. . . ,σk)∈ BCF(Q,P ) happenstohaveσ0equalto
thetrivialchain,i.e.|σ0|= 0,thenQ istheendpointofσ0soσ1 hastostartatQ∗.The
converseisalsotrue,ifσ1startsatQ∗,thenσ0hastobetrivial.Inthiscase(σ1,. . . ,σk)
isexactlythedata ofatethered brokenchainlinkingQ∗ to P .
Hencethetetheredbrokenchains(σ1,. . . ,σk)∈ tBCF(Q∗,P ) correspondpreciselyto
thebrokenchains(σ0,σ1,. . . ,σk)∈ BCF(Q,P ) whereσ0 isthetrivialchain.Thisway,
indiagram form,atetheredbrokenchainlinkingQ (orrather Q∗)toP looks like
Q Q∗ · · · a1n1 a2 0 · · · a2n2 . .. ak0 · · · aknk ∼ < < ∼ < < ∼ ∼ < < withak
nk ∼F P .Drawninthisform,thetotallengthofthetetheredbrokenchainisthe
totalnumberof < and ∼ symbols,where the initialQ∼ Q∗ adds thenecessary +1 in
comparisonwithDefinition 4.7.
Theorem 4.9canthusbereformulated as
Corollary 5.4 (Theorem 4.9 revisited). Let F be a saturated fusion system over a
fi-nitep-group S.The numbers of fixed pointsforeach irreducible F-stableset αP can be calculatedas (αP)Q=W SP∗ · σ=(σ0,...,σk)∈BCF(Q,P ) |σ0|=0 (−1)(σ)
Example5.5.LetS = D8andF = FS(A6) asbefore.WeshowedearlierthatcQ(αP)= 1
when Q = C21 and P = V42. Note that inthis case |WSP∗| =|WSQ| = 2 andthere is
onlyonebrokenchainfromC1
2 toV42 whichis C1 2 Z V2 4 ∼ <
Note thatthis isalsoatethered brokenchain.Sowehave|(αP)Q|=|WSP∗|· 1= 2 for
Q= C1
2 andP = V42.
IfwerepeatthesamecalculationforQ= C1
2 andP = D8,thenweobservethatthere
are 10 brokenchainsfrom C1
2 to D8 whichare C1 2 D8 C1 2 V41 D8 < < < C1 2 Z D8 ∼ < C1 2 V41 V1 4 D8 < ∼ < C21 Z V41 D8 ∼ < < C21 Z C4 D8 ∼ < < C21 Z V42 D8 ∼ < < C1 2 Z V1 4 V1 4 D8 ∼ < ∼ < C1 2 Z C4 C4 D8 ∼ < ∼ < C1 2 Z V2 4 V2 4 D8 ∼ < ∼ <
If we sum thesigns (−1)(σ) over all the broken chainsabove, and multiplyit with
|WSP∗|/|WSQ|,weget
cQ(αP) =1
2(1− 2 + 4 − 3) = 0.
Note thatifweonlyconsiderthetethered brokenchains,then weobtain
(αP)Q=W
SP∗(1− 3 + 3) = 1.
Note thatin the aboveexample someof the broken chainsnaturally canbe paired
secondrowcancelswiththebrokenchainsonthethirdrow.Inthenextsectionweprove
thatthebrokenchaincalculationsforcalculatingcQ(αP) and|(αP)Q| canbesimplified.
6. Computationalsimplifications
Inthis section, weshow thatcertaintypes ofbroken chainscanbe naturallypaired
withcertain other typesofbroken chainsinsuchaway thattheircontributions inthe
summationinTheorem 5.2canceleachother.Thisgivesamodifiedversionoftheformula
inTheorem 5.2wherewe onlyconsider brokenchainswhicharenotineithertype.We
startwithadefinitionofthese types.
Definition6.1.Letσ = (σ0,. . . ,σk) beabrokenchaininF withσi= (ai0,. . . ,aini).
Sup-posethatasubgroupai
j inthebrokenchainisS-conjugatetothechosenrepresentative
(ai
j)∗∈ P∗.Wesaythatsuchanaji isa∗-groupoftype 1 if0< j < ni,orifi= j = 0 and n0> 0.Wesaythataij is a∗-groupof type 2 ifj = ni and0≤ i< k.Intheremaining
caseswe eitherhavej = 0 and i> 0,in whichcase ai
j ∈ P∗ is always required, or we
havei= k andj = nk withaij as theverylastgroup.Ineitherof theselast cases,aij is
not a∗-group.
Indiagramformthetwo typesof∗-groupsareas follows:
. .. · · · (ai j)∗ · · · . .. < < or (a0 0)∗ · · · . .. < (Type 1) . .. · · · (ai j)∗ ai+10 · · · . .. < ∼ < or (a0 0)∗ a1 0 · · · . .. ∼ < (Type 2)
Ifabrokenchainσ containsatleastone∗-group,wesaythatσ issparklingof type 1
or 2 wherethetypeof σ isdeterminedbythetypeofthesmallest∗-groupin σ.A broken
chainis drab ifithasno∗-groupsatall.
Example6.2.ConsiderthelastcalculationinExample 5.5,whereQ= C1
2 andP = D8.
Thebrokenchainsonthesecondrowareallsparklingoftype 1.Morespecificallyinall
these,thesecondchainsinclude∗-groupsoftype 1 whichareV1
4,C4,andV42.Notealso
thatthe secondbroken chainonthefirstrowis asparklingbroken chainof thetype 1.
Thefourthchainonthefirstrowandallthechainsonthethirdrowaresparklingbroken
inan obviousway. Theonlydrab brokenchainsinthis examplearethefirst andthird
broken chainsonthefirstrow.
Proposition6.3.LetF beasaturatedfusionsystemoverafinitep-groupS.Incalculating
the coefficientscQ(αP) byTheorem 5.2, itissufficient toconsideronly thedrabbroken chains: cQ(αP) = |WS P∗| |WSQ| · σ∈BCF(Q,P ) σis drab (−1)(σ)
forQ,P ∈ P, whereP∗∼F P isfullynormalized.
Proof. ByTheorem 5.2wehave
cQ(αP) =|WSP ∗| |WSQ| · σ∈BCF(Q,P ) (−1)(σ)
for Q,P ∈ P, where P∗ ∼F P is fully normalized. For each subgroup R ≤ S we will
consider allthesparklingbrokenchainsthathaveR astheirsmallest∗-groupandlinks
Q to P .ForeachR we willshowthatthese brokenchainscanceleachother inthesum
above, leavingonly the drabbroken chains at theend. Inorder for R to be a∗-group
at all, R must be S-conjugateto the chosenrepresentative R∗ ∈ P∗. Wecantherefore
chooseans∈ S suchthatsR = R∗,andwelets befixedfortheremainderoftheproof.
Let σ∈ BCF(Q,P ) beabroken chainwith R as itssmallest∗-group. Suppose σ =
(. . . ,σ∗,. . .) where σ∗ is thechain containingR as a∗-group. If R is at theend of σ∗,
then σ istype 2,otherwiseσ istype 1.
If σ is type 1, then we write σ∗ = (A0,. . . ,Am−1,R,B1,. . . ,Bn) where n ≥ 1.
We can then conjugate the entire second part of the chain with s to get subgroups
Ci := sBi. These form a chain (R∗,C1,. . . ,Cn) which starts at R∗ ∈ P∗ and has
length at least 1 (see the illustration (6.1) below). We also have Cn ∼S Bn ∼F Bn∗,
so we can “break” σ∗ at R into two chains and get a legal broken chain σ :=
(. . . ,(A0,. . . ,Am−1,R),(R∗,C1,. . . ,Cn),. . .) wherewedon’tchangeanyotherpartof σ.
Thenewbrokenchainσ istype 2 withR asitssmallest∗-group.Sinceσhasoneextra
break comparedto σ,(σ)= (σ)+ 1.
Ifalternativelyσ hastype 2,wewriteσ∗= (A0,. . . ,Am−1,R) andlet(R∗,C1,. . . ,Cn)
be the chain of σ that follows σ∗ (such a chain exists since R is not the very last
group of σ). We conjugate every Ci with s from the right Bi := Cis, and they form
a chain (R,B1,. . . ,Bn) starting at R and satisfying Bn ∼S Cn ∼F Cn∗. We canthen
combine σ∗ with the Bi-chain to get a single chain, and a new broken chain σ :=
(. . . ,(A0,. . . ,Am−1,R,B1,. . . ,Bn),. . .) of type 1 with R as its smallest ∗-group. We
also have(σ)= (σ)− 1.
. .. A0 · · · Am−1 R B1 · · · Bn R∗ C1 · · · Cn . .. ∼ < < < < < < S ∼ S ∼ < < < ∼ S ∼ Type 1 Type 2 (6.1)
Becauseany twocorrespondingbroken chainshavelengthsthatdifferby1,theycancel
inthesumofTheorem 5.2. 2
Anotherway to reducethe numberof terms inthe sum of Theorem 5.2, is to limit
thesizes of the individual chainsina broken chain.This stems from thefact thatthe
usualMöbiusfunctionforsubgroupsofp-groupshasμ(A,B)= 0 unless B≤ NSA with
B/A elementaryabelian(see[7,Corollary 3.5],[8,Proposition 2.4]).
Proposition6.4.LetF beasaturatedfusionsystemoverafinitep-groupS.Incalculating
the coefficients cQ(αP) by Theorem 5.2, it is sufficient to consider only broken chains (σ0,. . . ,σk) where every σi = (ai0,. . . ,aini) has a
i ni ≤ NS(a i 0) with aini/a i 0 elementary abelian.Therefore, wehave
cQ(αP) = |WS P∗| |WSQ| · σ=((ai j)nij=0)ki=0∈BCF(Q,P ), s.t. each ai ni/a i 0is elm.ab. (−1)(σ)
forQ,P ∈ P, whereP∗∼FP is fullynormalized.
Proof. IntheproofofTheorem 5.2 weconsiderthesum
cQ(αP) = |WSP ∗| |WSQ| R0,R1,...,Rk∈P s.t. Rk∼FP∗, |Q|≤|R0|<|R1|<···<|Rk| (−1)kμ(Q, R0)μ R∗0, R1 μR∗1, R2 · · · μRk∗−1, Rk
A term of this sumis only nonzero ifQ R0 and Ri−1 Ri with elementary abelian
quotientsforalli.Hencethesumreduces to
cQ(αP) =|WSP ∗| |WSQ| R0,R1,...,Rk∈P s.t. Rk∼FP∗, |Q|≤|R0|<|R1|<···<|Rk|,
R0/Q and Ri/Ri−1are elm.ab.
(−1)kμ(Q, R0)μ R∗0, R1 μR∗1, R2 · · · μR∗k−1, Rk
As inthe proof of Theorem 5.2 we then replace each product of Möbius functions by
broken chainsandarriveattheformulaintheproposition. 2
Remark6.5. Sadlythetwo reductionsofPropositions 6.3 and 6.4cannot becombined,
as thatwould requirecanceling thesame brokenchainwith two different otherbroken
chains. To see this, let F = FD8(A6) be as in Example 3.2, where we showed that
αD8 = [S/D8].Letusshowthatifweexcludeboththesparklingbrokenchainsandthose
that violate the hypothesisof Proposition 6.4, then we would notbe ableto compute
thecoefficientoftheorbit[S/C1
2] inαD8 correctly.
AsitislistedinExample 5.5,thereareatotalof10brokenchainslinkingC1
2toD8.Of
these,only(C1
2 < D8) and(C21,Z < D8) aredrab,andofthose,onlythesecondwouldbe
countedinProposition 6.4.Thusthereisnochanceforcancelation,andtheintersections
of Propositions 6.3 and 6.4 wouldyield cC1
2(αD8)= 1/2, which is obviously false. The
issue isthatthere canbecancelation betweensparkling subgroupsand subgroupsthat
violatethehypothesisofProposition 6.4, sothatbycombiningbothconditionswemay
undercountthecancelationsneededintheproofofTheorem 5.2.
7. Anapplicationto characteristicbisets
InthissectionwedemonstratehowwecanuseTheorem 5.2togivestructuralresults
fortheminimal characteristicbisetassociatedto asaturatedfusionsystem.
Definition7.1.Weconsider(S,S)-bisets,i.e.finitesetsequippedwithbothaleftS-action
and aright S-action,andsuchthattheactionscommute.Thestructureofsuchabiset
X is equivalent to anaction ofS× S onX with (s1,s2).x= s1.x.(s2)−1,and for each
point x∈ X wespeakofthestabilizerStabS×S(x) asasubgroupofS× S.
An F-characteristic biset for a fusion system F on S is a biset Ω satisfying three
propertiesoriginallysuggestedbyLinckelmannandWebb:
(i) For every point ω ∈ Ω the stabilizer StabS×S(ω) has theform of agraph/twisted
diagonal Δ(P,ϕ) for some ϕ ∈ F(P,S) and P ≤ S, where the twisted diagonal Δ(P,ϕ)≤ S × S isdefinedas
Δ(P, ϕ) = ϕ(s), s s∈ P.
(ii) Ω is F-stable with respect to both S-actions. For bisets thatsatisfy property (i)
thisboils downto checkingthatthenumberoffixed pointssatisfy
ΩΔ(P,id)=ΩΔ(P,ϕ)=ΩΔ(ϕP,id) forallϕ∈ F(P,S) andP ≤ S.
(iii) Theprimep doesnotdivide|Ω|/|S| (whichisanintegerbecauseof(i)).Thisensures
In [9] it is shown that there exists a characteristic biset for F if and only if F is
saturated,anditisshownhowto reconstructF given anyF-characteristicbiset.In[5]
two of the authors of this paper give a parametrization of all the characteristic bisets
for agiven saturatedfusion systemF. In particular it isshown thatthere is aunique
minimal F-characteristic biset ΛF, and every other F-characteristic biset contains at
leastonecopyof ΛF.
Theorem 7.2. (See [5, Theorem 5.3 and Corollary 5.4].) Let F be a saturated fusion
systemon afinite p-group S,andconsider theproductfusion system F × F onS× S. AccordingtoProposition 2.2there isan irreducible(F × F)-stable(S× S)-set αΔ(S,id) corresponding tothediagonal Δ(S,id)≤ S × S.Denote this (S× S)-set or(S,S)-biset byΛF:= αΔ(S,id).
The biset ΛF is then F-characteristic, and every F-characteristic biset contains a copy of ΛF (up to isomorphism).Hence ΛF is the unique minimalcharacteristic biset forF.
InordertoapplyTheorem 5.2tostudyΛF weneedtofigureoutwhatbrokenchains
looklike inthecontextof bisetsand thefusionsystemF × F.
In a product fusion system the conjugation is defined coordinate-wise. Hence two
twisted diagonals Δ(P,ϕ) and Δ(P,ϕ) are conjugate in F × F if and only if there
are additionalisomorphismsψ,ρ∈ F such thatϕ = ψ◦ ϕ◦ ρ−1.Consequently, every
Δ(P,ϕ) withϕ ∈ F(P,S) isconjugate to Δ(P,id) which is conjugate to Δ(P,id) for
allP ∼F P . Inaddition the subgroups ofS× S that aresubconjugate to Δ(S,id) in
F × F areprecisely allthetwisteddiagonals Δ(P,ϕ) withϕ∈ F(P,S) and P ≤ S. To
study ΛF = αΔ(S,id) we thereforehave to consider broken chainswhere all the groups
aretwisteddiagonalscomingfrommapsinF.
Two twisted diagonals satisfy Δ(Q,ψ) ≤ Δ(P,ϕ) exactly when ϕ extends ψ, i.e.
Q ≤ P and ψ = ϕ|Q. Every (F × F)-conjugacy class of twisted diagonals contains a
fully normalizedrepresentative on theform Δ(P∗,id) where P∗ is fully F-normalized,
suppose for Theorem 5.2 that we have chosen such at fully normalized representative
Δ(P∗,id) for each conjugacy class. The broken chains that we consider are chains of
inclusionsconnectedby(F × F)-conjugations.
• Every chain of inclusionsΔ(P1,ϕ1) ≤ · · · ≤ Δ(Pk,ϕk) is asequence of extensions
withϕi= ϕk|Pi.
• Everychain(exceptforthe0’thchain)startswithadiagonaloftheformΔ(P∗,id)
where P∗ afullynormalizedrepresentativefortheF-conjugacyclass.
Withthis insight we cannow apply Theorem 5.2 and relateΛF to the largest normal
subgroup in F. Here normality is in the sense of [1, Definition 4.3] where P ≤ S is
normalinF ifeveryhomomorphismϕ∈ F(Q,R) extendstosomeϕ ∈ F(QP,RP ) with
