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P ojec i e modules and in olu ions
John Mu ay
Ma hema ics Depa men , Na ional Uni e si y o I eland, Maynoo h, Co. Kilda e, I eland
Recei ed 30 Ma ch 2005
Communica ed by Michel B oué
Abs ac
Le Gbe a ini e g oup, and le Ω:= { ∈G| 2=1}.ThenΩis a G-se unde conjuga ion. Le
kbe an algeb aically closed ield o cha ac e is ic 2. I is shown ha each p ojec i e indecomposable
summand o he G-pe mu a ion module kΩ is i educible and sel -dual, whence i belongs o a eal
2-block o de ec ze o. This, oge he wi h he ac ha each i educible kG-module ha belongs o
a eal 2-block o de ec ze o occu s wi h mul iplici y 1 as a di ec summand o kΩ, es ablishes a
bijec ion be ween he p ojec i e componen s o kΩ and he eal 2-blocks o Go de ec ze o.
2005 Published by Else ie Inc.
Keywo ds: In olu ions; Blocks o de ec ze o; G een co espondence; Bu y–Ca lson–Puig heo em
Le Gbe a ini e g oup, wi h iden i y elemen e, and le Ω:= { ∈G| 2=e}. Then
Ωis a G-se unde conjuga ion. In his no e we desc ibe he p ojec i e componen s o he
pe mu a ion module kΩ, whe e kis an algeb aically closed ield o cha ac e is ic 2. By a
p ojec i e componen we mean an indecomposable di ec summand o kΩ ha is also a
di ec summand o a ee kG-module. We show ha all such componen s a e i educible,
sel -dual and occu wi h mul iplici y 1.
This gi es an al e na i e p oo o Rema k (2) on p. 254 o [5], and s eng hens Co ol-
la ies 3 h ough 7 o ha pape . In addi ion, we can gi e he ollowing quick p oo o
P oposi ion 8 in [5]:
E-mail add esses: [email p o ec ed], [email p o ec ed].ie.
0021-8693/$ – see on ma e 2005 Published by Else ie Inc.
doi:10.1016/j.jalgeb a.2005.05.032
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Co olla y 1. Suppose ha His a s ongly embedded subg oup o G. Then kH↑G∼
=
kG⊕[
s
i=1Pi]whe e s⩾0and he Pia e pai wise nonisomo phic sel -dual p ojec i e
i educible kG-modules.
P oo . Tha His s ongly embedded means ha |H|is e en and |H∩Hg|is odd, o
each g∈G H.Le ∈Hbe an in olu ion. Then clea ly CG( ) ⩽H.SokH↑Gis isomo -
phic o a submodule o (kCG( ))↑G. Mackey’s heo em implies ha e e y componen o
kH↑G, o he han kG, is a p ojec i e kG-module. Being p ojec i e, hese modules mus be
componen s o (kCG( ))↑G. The esul now ollows om Theo em 8. 2
Conside he w ea h p oduc GΣo Gwi h a cyclic g oup Σo o de 2. He e Σis
gene a ed by an in olu ion σand GΣis isomo phic o he semidi ec p oduc o he base
g oup G×Gby Σ. The conjuga ion ac ion o σon G×Gis gi en by (g1,g
2)σ=(g2,g
1),
o all g1,g
2∈G. The elemen s o GΣwill be w i en (g1,g
2),(g1,g
2)σ o σ.
We shall exploi he ac ha kG is a kG Σ-module. Fo , as is well known, kG is an
k(G ×G)-module ia: x·(g1,g
2):= g−1
1xg2, o each x∈kG, and g1,g
2∈G. The ac ion
o Σon kG is induced by he pe mu a ion ac ion o σon he dis inguished basis Go kG:
gσ:= g−1, o each g∈G. Clea ly σac s as an in olu a y k-algeb a an i-au omo phism
o kG. I ollows ha he ac ions o G×Gand Σon kG a e compa ible wi h he g oup
ela ions in GΣ.
Byablock o kG, o a 2-block o G, we mean an indecomposable k-algeb a di ec sum-
mand o kG. Each block has associa ed o i a p imi i e idempo en in Z(kG), a B aue
equi alence class o cha ac e s o i educible kG-modules and a B aue equi alence class,
modulo 2, o o dina y i educible cha ac e s o G. A block has de ec ze o i i is a simple
k-algeb a, and is eal i i con ains he complex conjuga es o i s o dina y i educible cha -
ac e s. Theo em 8 es ablishes a bijec ion be ween he eal 2-blocks o G ha ha e de ec
ze o and he p ojec i e componen s o kΩ.
We could equally well wo k o e a comple e disc e e alua ion ing Ro cha ac e is-
ic 0, whose ield o ac ions Fis algeb aically closed, and whose esidue ield R/J(R)
is k.SoweuseO o indica e ei he o he commu a i e ings ko R.
All ou modules a e igh -modules. We deno e he i ial OG-module by OG.I Mis an
OG-module, we use M↓H o deno e he es ic ion o M o H.I His a subg oup o Gand
Nis an OH-module, we use N↑G o deno e he induc ion o N o G. Whene e g∈G,we
w i e g o (g, g) ∈G×G, and we se X:= {x|x∈X}, o each X⊂G. O he no a ion
and concep s can be ound in a s anda d ex book on modula ep esen a ion heo y, such
as [1] o [4].
I Bis a block o OG, hen so oo is Bo={xσ|x∈B}. We call Ba eal block i
B=Bo. Ou i s esul desc ibes he componen s o OGas OGΣ-module.
Lemma 2. The e is an indecomposable decomposi ion o OGas OGΣ-module:
OG=B1⊕···⊕B ⊕B +1+Bo
+1⊕···⊕B +s+Bo
+s+1.
He e B1,...,B
a e he eal 2-blocks and B +1,Bo
+1,...,B
+s,Bo
+sa e he non eal
2-blocks o G.
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P oo . This ollows om he well-known indecomposable decomposi ion o OG,asan
O(G ×G)-module, in o a di ec sum o i s blocks, and he ac ha Bσ
i=Bi o i=
1,..., , and Bσ
+j=Bo
+j o j=1,...,s.2
An ob ious bu use ul ac is ha OGis a pe mu a ion module:
Lemma3.The OGΣ-module OGis isomo phic o hepe mu a ionmodule (OG×Σ)↑GΣ.
P oo . The elemen s o G o m a GΣ-in a ian basis o OG. Mo eo e i g1,g
2∈G,
hen g2=g1·(g1,g
2).SoGis a ansi i e GΣ-se . The s abilize o e∈OGin GΣis
G×Σ. The lemma ollows om hese ac s. 2
Le Cbe a conjugacy class o G. Se Co:= {c∈G|c−1∈C}. Then Cois also a
conjugacy class o G, and C∪Cocan be ega ded as an o bi o G×Σon he GΣ-
se G. As such, he co esponding pe mu a ion module O(C ∪Co)is a OG×Σ-di ec
summand o OG.I C=Co, we call Ca eal class o G. In his case o each c∈C he e
exis s x∈Gsuch ha cx=c−1. The poin s abilize o cin G×Σis CG(c)xσ.So
OC∼
=(OCG(c)xσ)↑G×Σ.
I C= Co, we call Ca non eal class o G. In his case he poin s abilize o c∈C∪Coin
G×Σis CG(c).So
O(C ∪Co)∼
=(OCG(c))↑G×Σ.
Suppose now ha he eal classes a e C1,...,C
and ha he non eal classes a e
C +1,Co
+1,...,C
+u,Co
+u. Then we ha e:
Lemma 4. The e is a decomposi ion o OGas an OG×Σ-pe mu a ion module:
OG=OC1⊕···⊕OC ⊕OC +1∪Co
+1⊕···⊕OC +u∪Co
+u+1.
P oo . This ollows om Lemma 3 and he discussion abo e. 2
By a quasi-pe mu a ion module we mean a di ec summand o a pe mu a ion module.
Ou nex esul is Lemma 9.7 o [1]. We include a p oo o he con enience o he eade .
Lemma 5. Le Mbe an indecomposable quasi-pe mu a ion OG-module and suppose ha
His a subg oup o Gsuch ha M↓His indecomposable. Then he e is a e ex Vo M
such ha V∩His a e ex o M↓H.I His a no mal subg oup o G, hen his is ue o
all e ices o M.
P oo . Le Ubea e exo M.AsOU|M↓Uwe ha e OU∩H|(M↓H)↓U∩H.Bu U∩H
is a e ex o OU∩H. So Mackey’s heo em implies ha he e exis s a e ex Wo M↓H
such ha U∩H⩽W.
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As M↓His a componen o he es ic ion o M o H, Mackey’s heo em shows ha
he e exis s g∈Gsuch ha W⩽Ug∩H.NowUgis a e ex o M. So by he p e ious
pa ag aph, and he uniqueness o e ices o M↓Hup o H-conjugacy, he e exis s h∈H
such ha Ug∩H⩽Wh. Compa ing ca dinali ies, we see ha W=Ug∩H.SoUg∩H
is a e ex o M↓H.
Suppose ha His a no mal subg oup o G. Then U∩H⩽Wand W=Ug∩H=
(U ∩H)gimply ha U∩H=W.2
R. B aue showed how o associa e o each block o OGaG-conjugacy class o
2-subg oups, i s so-called de ec g oups. I is known ha a block has de ec ze o i and
only i i s de ec g oups a e all i ial. J.A. G een showed how o associa e o each inde-
composable OG-module a G-conjugacy class o 2-subg oups, i s so-called e ices. He
also showed how o iden i y he de ec g oups o a block using i s e ices as an indecom-
posable O(G ×G)-module.
Co olla y 6. Le Bbe a block o OGand le Dbe a de ec g oup o B.I Bis no eal
hen Dis a e ex o B+Bo,asOGΣ-module. I Bis eal, hen he e exis s x∈NG(D),
wi h x2∈D, such ha Dxσ is a e ex o B,asOGΣ-module. In pa icula , Σis a
e ex o B+Boi and only i Bis a eal 2-block o G ha has de ec ze o.
P oo . J.A. G een showed in [2] ha Dis a e ex o B, when Bis ega ded as an
indecomposable O(G ×G)-module. Suppose i s ha Bis no eal. Then B+Bo=
(B↓G×G)↑GΣ, o ins ance by Co olla y 8.3 o [1]. I ollows ha B+Bohas e ex
D, as an indecomposable OGΣ-module.
Suppose hen ha B=B+Bois eal. Lemma 3 shows ha Bis G×Σ-p ojec i e. So
we may choose a e ex Vo Bsuch ha V⩽G×Σ. Mo eo e , Bis a quasi-pe mu a ion
OGΣ-module, and i s es ic ion o he no mal subg oup G×Gis indecomposable.
Lemma 5 hen implies ha V∩(G ×G) =V∩Gis a e ex o B↓G×G. So by G een’s
esul , we may choose Dso ha V∩G=D.NowG×Ghas index 2 in GΣ. So G een’s
indecomposabili y heo em, and he ac ha B↓G×Gis indecomposable, implies ha V⊆
(G ×G). I ollows ha he e exis s x∈NG(D), wi h x2∈D, such ha V=Dxσ .
I Bhas de ec ze o, hen D=e.Sox2=e. In his case, xσ =Σ(e,x) is GΣ-
conjuga e o Σ.SoΣis a e ex o B. Con e sely, suppose ha Σis a e ex o B+Bo.
The i s pa ag aph shows ha Bis a eal block o G. Mo eo e Bhas de ec ze o, as
Σ∩G=e.2
We quo e he ollowing esul o Bu y, Ca lson and Puig [4, 4.4.6] on he G een co e-
spondence:
Lemma 7. Le V⩽H⩽Gbe such ha Vis a p-g oup and NG(V ) ⩽H. Le deno e he
G een co espondence wi h espec o (G,V,H). Suppose ha Mis an indecomposable
OG-module such ha M↓Hhas a componen Nwi h e ex V. Then Vis a e ex o M
and N= (M).
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We can now p o e ou main esul . Pa (ii) is Rema k (2) on p. 254 o [5], bu ou p oo
is independen o he p oo gi en he e.
Theo em 8.
(i) Le ∈G, wi h 2=e. Suppose ha Pis an indecomposable p ojec i e di ec sum-
mand o (OCG( ))↑G. Then Pis i educible and sel -dual and occu s wi h mul iplici y
1as a componen o (OCG( ))↑G. In pa icula Pbelongs o a eal 2-block o G ha
has de ec ze o.
(ii) Suppose ha Mis a p ojec i e indecomposable OG-module ha belongs o a eal
2-block o G ha has de ec ze o. Then he e exis s s∈G, wi h s2=e, such ha M
is a componen o (OCG(s))↑G. Mo eo e , sis uniquely de e mined up o conjugacy
in G.
P oo . I =e hen P=OG.SoPis i educible and sel -dual. The assump ion ha Pis
p ojec i e and he ac ha dimO(P ) =1 implies ha |G|is odd. So all blocks o OG,in
pa icula he one con aining P, ha e de ec ze o.
Now suppose ha = e.Le Tbe he conjugacy class o G ha con ains . The pe mu-
a ion module OTis a di ec summand o he es ic ion o OG o G×Σ.Rega dPas an
OG-module. Le I(P) be he in la ion o his module o G×Σ. Then I(P) is a compo-
nen o OT.AsΣis con ained in he ke nel o I(P), and Pis a p ojec i e OG-module, i
ollows ha I(P)has e ex Σas an indecomposable OG×Σ-module.
By Lemma 2, and he K ull–Schmid heo em, he e exis s a 2-block Bo Gsuch ha
I(P) is a componen o he es ic ion (B +Bo)↓G×Σ. An easy compu a ion shows ha
NGΣ(Σ) =G×Σ. I hen ollows om Lemma 7 ha (B +Bo)has e ex Σand also
ha I(P)is he G een co esponden o (B +Bo)wi h espec o (G Σ,Σ,G ×Σ).We
conclude om Co olla y 6 ha Bis a eal 2-block o G ha has de ec ze o.
Le ˆ
Bbe he 2-block o GΣ ha con ains B. Then ˆ
Bis eal and has de ec g oup Σ.
Le ˆ
Abe he B aue co esponden o ˆ
B. Then ˆ
Ais a eal 2-block o G×Σ ha has
de ec g oup Σ.Now ˆ
A=A⊗OΣ, whe e Ais a eal 2-block o OG ha has de ec
ze o. In pa icula Ahas a unique indecomposable module, and his module is p ojec i e,
i educible and sel -dual. Co olla y 14.4 o [1] implies ha I(P) belongs o ˆ
A.SoP
belongs o A. We conclude ha Pis i educible and sel -dual and belongs o a eal 2-block
o G ha has de ec ze o.
Now Boccu s wi h mul iplici y 1 as a componen o OG, and I(P) is he G een
co esponden o Bwi h espec o (G Σ,Σ,G ×Σ).SoI(P) has mul iplici y 1 as
a componen o he es ic ion o OG o G×Σ. I ollows ha Poccu s wi h mul iplici y
1 as a componen o (OCG( ))↑G, and wi h mul iplici y 0 as a componen o (OCG( ))↑G,
o ∈Gwi h 2=e,bu no G-conjuga e o . This comple es he p oo o pa (i).
Le Rbe a eal 2-block o G ha has de ec ze o. Then Rhas e ex Σas indecompos-
able OGΣ-module. So i s G een co esponden (R), wi h espec o (GΣ,Σ,G×Σ),
is a componen o he es ic ion o OG o G×Σ ha has e ex Σ. Lemma 4 and he
K ull–Schmid heo em imply ha (R)is isomo phic o a componen o O(C ∪Co), o
some conjugacy class Co G.NowΣis a cen al subg oup o G×Σ.SoΣmus be a
subg oup o he poin s abilize o C∪Coin G×Σ. I ollows ha s2=e, o each s∈C.

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Le Ndeno e he es ic ion o (R) o G, and conside Nas an OG-module. We ha e
jus shown ha Nis a componen o (OCG(s))↑G. A guing as be o e, we see ha Nis an
indecomposable p ojec i e OG-module ha belongs o a eal 2-block o G ha has de ec
ze o.
The las pa ag aph es ablishes an injec i e map be ween he eal 2-blocks o G ha ha e
de ec ze o and ce ain p ojec i e componen s o OΩ. As each block o de ec ze o con-
ains a single i educible OG-module, his map mus be on o. I ollows ha he module M
in he s a emen o he heo em is a componen o some pe mu a ion module (OCG(s))↑G,
whe e s∈Gand s2=e. The ac ha sis de e mined up o G-conjugacy now ollows om
he las s a emen o he p oo o pa (i). This comple es he p oo o pa (ii). 2
I is possible o simpli y he abo e p oo by showing ha i Bis a eal 2-block o G ha
has de ec ze o, hen i s G een co esponden , wi h espec o (G Σ,Σ,G ×Σ) is MF ,
whe e MF is he F obenius conjuga e o he unique i educible OG-module ha belongs
o B.
Suppose ha Ris a comple e disc e e alua ion ing and ha Lis an RCG( )-module,
whe e Lhas R- ank 1 and O2(CG( )) ac s i ially on L. Then he 2-modula educ ion o
Lis he i ial kCG( )-module, al hough Lis no necessa ily he i ial RCG( )-module.
Now each p ojec i e i educible kG-module li s o a p ojec i e i educible RG-module.
So he conclusions o pa (i) o he abo e heo em apply o L↑G: all o i s p ojec i e com-
ponen s a e i educible and sel -dual. We hank he e e ee o poin ing ou his ex ension
o ou esul .
The p oo o Theo em 8 hin s a he ac ha we ha e some 2-local con ol o e all he
componen s o (OCG( ))↑G. The in es iga ion o special p ope ies o such componen s is
con inued in [3].
Co olla y 9. Le Ω={ ∈G| 2=e}. Then he e is a bijec ion be ween he eal 2-blocks
o G ha ha e de ec ze o and he p ojec i e componen s o OΩ.
He e is a sample applica ion. I was sugges ed o me by G.R. Robinson.
Co olla y 10. Le n⩾1and le be an in olu ion in he symme ic g oup Σn.I
n=m(m +1)/2is a iangula numbe , and is a p oduc o (m2+1)/4commu ing
ansposi ions, hen he e is a single p ojec i e i educible OΣn-module, and his module
is he unique p ojec i e componen o (OCΣn( ))↑Σn. Fo all o he alues o no noncon-
juga e in olu ions , he modules (OCΣn( ))↑Σna e p ojec i e ee.
P oo . We gi e a p oo o he ollowing esul in [3, Co olla y 8.4]: Le Gbe a ini e g oup,
le Bbe a eal 2-block o Go de ec ze o, and le χbe he unique i educible cha ac e
in B. Then he e exis s a 2- egula conjugacy class Co Gsuch ha C=Co,|CG(c)|is
odd, o c∈C, and χ(c) is nonze o, modulo a p ime ideal con aining 2. Mo eo e , he e
exis s an in olu ion ∈Gsuch ha c =c−1, and o his we ha e χCG( ),1CG( )=1.
The exis ence o was shown in [5]. The iden i ica ion o using he class Cwas i s
shown by R. Gow (in unpublished wo k).
ARTICLE IN PRESS
UNCORRECTED PROOF
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Suppose ha (OCΣn( ))↑Σnhas a p ojec i ecomponen .Then Σnhas a 2-block o de ec
ze o, by Theo em8. The 2-blocks o Σna e indexed by iangula pa i ions µ=[m, m−1,
...,2,1], whe e m anges o e hose na u al numbe s o which n−m(m +1)/2 is e en.
Mo eo e , he 2-block co esponding o µhas de ec ze o i and only i n=m(m +1)/2.
In pa icula , we can assume ha n=m(m +1)/2, o some m⩾1.
Le Bbe he unique 2-block o Σn ha has de ec ze o, le χbe he unique i educible
cha ac e in Band le g∈Σnha e cycle ype λ=[2m−1,2m−5,...]. Then |CΣn(g)|
is odd. As he pa s o λa e he “diagonal hookleng hs” o µ, he Mu naghan–Nakayama
o mula shows ha χ(g)=1. Now λhas (m −1)/2nonze o pa s. So gis in e ed by an
in olu ion ha is a p oduc o (n −(m −1)/2)/2=(m2+1)/4commu ing anspo-
si ions. I ollows om Theo em 8 and he p e ious pa ag aph ha he unique i educible
p ojec i e B-module occu s wi h mul iplici y 1 as a componen o (OCΣn( ))↑Σn.Thelas
s a emen o he co olla y now ollows om Theo em 8. 2
Re e ences
[1] J.L. Alpe in, Local Rep esen a ion Theo y, Camb idge S ud. Ad . Ma h., ol. 11, 1986.
[2] J.A. G een, Blocks o modula ep esen a ions, Ma h. Z. 79 (1962) 100–115.
[3] J. Mu ay, Ex ended de ec g oups and ex ended e ices, Osaka J. Ma h, in p ess.
[4] H. Nagao, Y. Tsushima, Rep esen a ions o Fini e G oups, Academic P ess, 1989.
[5] G.R. Robinson, The F obenius–Schu indica o and p ojec i e modules, J. Algeb a 126 (1989) 252–257.