Monomial G-posets and their Lefschetz invariants

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Journal

of

Algebra

www.elsevier.com/locate/jalgebra

Monomial

G-posets

and

their

Lefschetz

invariants

Serge Bouca_{, Hatice Mutlu}b,∗

a_{CNRS-LAMFA,UniversitédePicardie- JulesVerne,33rueStLeu,} 80039 - Amiens, France

b

DepartmentofMathematics,BilkentUniversity,06800- Bilkent,Ankara, Turkey

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received15August2018 Availableonline14March2019 CommunicatedbyMarkus Linckelmann MSC: 06A11 19A22 20J15 Keywords: Burnsidering Monomial Tensorinduction Lefschetzinvariant

Let G be a finite group, and C be an abelian group. We introducethenotionsofC-monomialG-setsandC-monomial G-posets,andstatesomeoftheircategoricalproperties.This gives in particular a new description of the C-monomial BurnsideringBC(G).WealsointroduceLefschetzinvariants

of C-monomial G-posets, which are elements of BC(G).

Theseinvariantsallowforadefinitionofageneralizedtensor induction multiplicative map TU,λ : BC(G) → BC(H)

associated to any C-monomial (G,H)-biset (U,λ), which inturn givesa group homomorphismBC(G)× → BC(H)×

betweentheunitgroupsofC-monomialBurnsiderings. ©2019ElsevierInc.Allrightsreserved.

1. Introduction

LetG beafinitegroup,andC beanabeliangroup.Inthiswork,wefirstintroducethe notionofC-monomialG-set:thisisapair(X,l) consistingofafiniteG-setX,together withafunctorfromthetransportercategoryX of X,tothe groupoid•Cwithoneobject

andautomorphismgroupC.TheC-monomialG-setsformacategoryCM G-set,andwe

show that it is equivalent to the category CF G-set of C-fibred G-sets considered by

* Correspondingauthor.

E-mailaddresses:serge.bouc@u-picardie.fr(S. Bouc),hatice.mutlu@bilkent.edu.tr(H. Mutlu). https://doi.org/10.1016/j.jalgebra.2019.02.036

Barker ([1]). Inparticular, the C-monomial Burnside ring BC(G) introduced by Dress

([5])isisomorphictotheGrothendieckringofthecategoryCM G-set.

WeextendthesedefinitionstothenotionofC-monomialG-poset:thisisapair(X,l) consisting ofafiniteG-posetX,andafunctorl fromthetransportercategoryX to •C.

We associate to each such pair (X,l) aLefschetz invariant Λ(X,l) lying inBC(G). We

showthatanyelementofBC(G) isequaltotheLefschetzinvariantofsome(nonunique)

C-monomialG-poset.

Wealso introducethecategoryCM G-poset ofC-monomialG-posets,and showthat

therearenaturalfunctorsofinduction Ind_HG :CM H-poset→CM G-poset andof

restric-tion ResG_H:CM G-poset→CM H-poset,wheneverH isasubgroupofG.Thesefunctors

are compatiblewiththeconstructionofLefschetz invariants.

We extendseveralclassicalpropertiesoftheLefschetz invariantsof G-posetsto Lef-schetz invariantsofC-monomial G-posets (theclassicalcasebeing thecasewhere C is

trivial).

Wenextturnto theconstructionofgeneralizedtensor induction functors

TU,λ:CM G-poset→CM H-poset

associated,forarbitraryfinitegroupsG andH,toanyC-monomial(G,H)-biset(U,λ).

Weshow thatthesefunctors inducewelldefinedtensorinductionmaps

TU,λ: BC(G)→ BC(H),

which arenotadditiveingeneral,butmultiplicativeandpreserveidentity elements.In particular,wegetinducedgrouphomomorphismsbetweenthecorrespondingunitgroups of monomial Burnside rings,similar to those obtainedby Carman([4]) for otherusual representation rings.

Weshowmoreoverthatunderanadditionalassumption,thesetensorinduction func-tors andtheirassociated tensorinductionmaps arewellbehaved forcomposition.This yieldstoa(partial)fibredbisetfunctorstructureonthegroupofunitsofthemonomial Burnside ring.

2. ThemonomialBurnside ring

LetG beafinitegroupandC beanabeliangroupwhichisnotedmultiplicatively.We denote byG-set thecategory of finiteG-sets (withG-equivariantmaps asmorphisms), and B(G) theusualBurnsideringofG,i.e.theGrothendieckringofG-set forrelations given bydisjointuniondecompositionsoffinite G-sets.

2.1. Thecategory of C-fibredG-sets

AC-fibredG-set isdefinedtobeaC-free(C×G)-setwithfinitelymanyC-orbits.Let

maps.ThecoproductofC-fibredG-setsX,Y istheircoproductX Y assets,withthe obvious(C× G)-action.IfX andY are C-fibred G-sets,there isaC-actiononX× Y

defined by c(x,y) = (cx,c−1y) for any c ∈ C and (x,y) ∈ X × Y . The C-orbit of an element(x,y) ofX× Y isdenotedbyx⊗ y andthesetofC-orbitsisdenotedbyX⊗ Y .

MoreoverC× G actsonX⊗ Y by

(c, g)(x⊗ y) = cgx ⊗ gy

for any (c,g) ∈ C × G andx⊗ y ∈ X ⊗ Y . One checks easily thatX ⊗ Y is again a

C-fibredG-set, calledthetensorproductofX andY .

We denote the isomorphism class of a C-fibred G-set X by [X]. The C-monomial Burnside ring BC(G),introduced by Dress ([5]), is definedas the Grothendieckgroup

ofthecategory ofC-fibredG-sets, forrelationsgivenby[X]+ [Y ]= [X Y ].Thering structureofBC(G) isinducedby[X]· [Y ]= [X⊗ Y ].Theidentity elementisthesetC

withtrivialG-actionandthezeroelementistheemptyset.IfC istrivialwerecoverthe ordinaryBurnside ringofthegroupG.

GivenaC-fibredG-setX,wedenotethesetofC-orbitsonX byC\X.Thegroup G acts onC\X, and X is (C× G)-transitive ifand only if C\X isG-transitive. If C\X

is transitiveas aG-set it isisomorphic to G/U forsomeU ≤ G.There exists agroup homomorphism μ: U → C suchthat ifU isthe stabilizer of the orbitCx,then ax=

μ(a)x foralla∈ U.Sincethestabilizer(C× G)x ofx inC× G isequalto

(C× G)x={



μ(a)−1, a| a ∈ U},

theC-fibredG-setX isdetermineduptoisomorphismbythesubgroupU andμ.

Conversely,letU beasubgroupofG,andμ: U→ C beagrouphomomorphism.Then wesetUμ={



μ(a)−1,a| a∈ U},anddenoteby[U,μ]GtheC-fibredG-set(C×G)/Uμ.

Thepair (U,μ) iscalled aC-subcharacter of G. Wedenote the set of C-subcharacters

by ch(G).ThegroupG actson ch(G) byconjugation.TheG-set ch(G) isaposetwith therelation≤ definedby

(U, μ)≤ (V, ν) ⇔ U ≤ V and ResV_Uν = μ

forany(U,μ) and(V,ν) in ch(G). Asanabeliangroupwe have

BC(G) =



(U,μ)∈Gch(G)

Z[U, μ]G

where(V,ν) runsoverG-representativesoftheC-subcharactersofG,detailscanbeseen in [1].

2.2. Thecategory of C-monomialG-sets

LetG beafinite groupandC beanabeliangroup.GivenaG-setX,weconsider its transportercategoryX whose objectsaretheelementsofX andgivenx,y inX theset of morphismsfromx toy is

Hom_X(x, y) ={g ∈ G | gx = y}.

Let•C denotethecategorywithoneobjectwheremorphismsaretheelementsofC and

compositionismultiplication inC.Nowwedefine C-monomialG-setsasfollows.

Definition 1.A C-monomial G-set isa pair(X,l) consistingof a finite G-set X and a functor l: X → •C.

In other words, foreach x,y ∈ X andg ∈ G such thatgx= y, wehave anelement l(g,x,y) ofC,withthepropertythatl(h,y,z)l(g,x,y)= l(hg,x,z) ifh∈ G andhy = z,

and l(1,x,x)= 1 foranyx∈ X.

Let (X,l) and (Y,m) be C-monomial G-sets. If f : X → Y is amap of G-sets, we slightly abuse notation and also denote by f : X → Y the obvious functor induced by f . Now amap(f,λ): (X,l)→ (Y,m) of C-monomialG-sets is apairconsisting of amap f : X → Y of G-sets andanaturaltransformation λ: l→ m◦ f. Wedenote by

CM G-set thecategory whoseobjects areC-monomial G-sets,morphisms arethemaps

of C-monomialG-sets,andcompositionistheobviousone.

Let (X,l) and (X,l) be C-monomial G-sets. We define the disjoint union of

C-monomial G-sets as (X,l) (X,l) = (X X,l l) where X X is the disjoint unionofG-setsand

l l: X X→ •C

is thefunctor suchthat

(l l)(g, z1, z2) =



l(g, z1, z2) z1, z2∈ X

l(g, z1, z2) z1, z2∈ X

forany z1,z2∈ X  X suchthatgz1= z2 forsomeg∈ G.

The product of C-monomial G-sets (X,l), (X,l) is defined to be (X × X,l× l) where X× XistheproductofG-setsandl× l: X× Y → •Cisthefunctordefinedby

(l× l)g, (x, x), (y, y)= l(g, x, y)l(g, x, y) forg∈ G and(x,x),(y,y)∈ X × X suchthatg(x,x)= (y,y).

Our goal is to show that the categories CM G-set and CF G-set are equivalent. For

this, wedefineafunctorF :CM G-set→CF G-set asfollows:givenaC-monomialG-set

F (X, l) = C×lX,

which is the direct product C × X endowed with the (C × G)-action defined by (k,g)(c,x)=kcl(g,x,gx),gxforany(k,g)∈ C × G and (c,x)∈ C × X.

Givenamap(f,λ): (X,l)→ (Y,m) of C-monomialG-sets,wedefine

F (f, λ) : C×lX → C ×mY

byF (f,λ)(c,x)= (cλx,f (x)) for any (c,x)∈ C ×lX.ThenF (f,λ) isa(C× G)-map:

indeed,given(k,g)∈ C × G and(c,x)∈ C × X,wehave (k, g)F (f, λ)(c, x) = (k, g)cλx, f (x)  =kcλxm(g, f (x), f (gx)), f (gx)  =kcλgxl(g, x, gx), f (gx)  = F (f, λ)kcl(g, x, gx), gx = F (f, λ)(k, g)(c, x).

ItisclearthatF :CM G-set→CF G-set is afunctor.

Lemma 2.Let C be an abelian group and G be a finite group. Then the above functor F :CM G-set→CF G-set isan equivalenceofcategories.

Proof. We provethat F is fully faithfuland essentially surjective. Firstwe show that

F is essentially surjective. Given aC-fibred G-set X, letC\X be the set of C-orbits.

Clearly C\X is aG-set. Wedefine a functorl : C\X → •C. LetCx, Cy ∈ C\X such

thatCgx= Cy for some g ∈ G. Then there exists auniquec ∈ C suchthat gx= cy. Wesetl(g,Cx,Cy)= c.WehaveF (C\X,l)= C×l(C\X).Now chooseaset[C\X] of

G-representatives ofthe G-actionon C\X. Then forany x∈ X, there exists a unique

Cσx ∈ [C\X] suchthatx∈ Cσx. Since X isC-free,there exists auniquecx ∈ C such

thatx= cxσx.Wedefinea(C×G)-mapf : X→ C×l(C\X) suchthatf (x)= (cx,Cσx).

Then

(c, g)f (x) = (c, g)(cx, Cσx) = (cxcl(g, Cσx, Cgσx), Cgσx) = (cxc, Cgσx)

= (ccgx, Cgσx) = f



(c, g)x.

Sof isa(C× G)-mapandclearlyanisomorphism.Thus,F isessentiallysurjective. Let(X,l) and(Y,m) beC-monomialG-sets.Weneedtoshow thatthemap

F : Hom(X, l), (Y, m)→ HomF (X, l), F (Y, m)

inducedbyF issurjectiveand injective.Letϕ: C×lX → C ×mY bea(C× G)-map.

Given(1,x)∈ C×lX,letϕ(1,x)= (cx,zx) for(cx,zx)∈ C×Y .Sinceϕ isa(C×G)-map,

ϕ(c, x) = (ccx, zx)

and

ϕ(1, gx) =cxm(g, zx, gzx)l−1(g, x, gx), gzx

 forany c∈ C and g∈ G. Wedefine amap

(f, λ) : (X, l)→ (Y, m)

suchthatf : X→ Y isdefinedbyf (x)= zxandλ: l→ m◦ f isdefinedbyλx= cxfor

any x∈ X.Clearly, f isaG-setmap.Letx∈ X andg∈ G.Then mg, f (x), f (gx)λx= m  g, f (x), f (gx)cx= cxm  g, f (x), f (gx)l−1(g, x, gx)l(g, x, gx) = l(g, x, gx)cgx= l(g, x, gx)λgx.

So λ: l→ m◦ f is anaturaltransformation and(f,λ) isamap ofC-monomialG-sets.

Thus, F (f,λ)= ϕ andF issurjective.Theinjectivityisclear,so F isfullyfaithful. 2 Proposition 3.Let G bea finite group. Then BC(G) is isomorphic to theGrothendieck

ring of thecategory CM G-set,forrelationsgiven bydecomposition intodisjointunions

of C-monomialG-sets andmultiplication inducedbyproductof C-monomialG-sets.

Proof. WeletB1

C(G) denotetheGrothendieckringofthecategoryCM G-set.The

equiv-alence F :CM G-set→CF G-set induces abijection  F : B_C1(G)→ BC(G) suchthat  F (X, l)= [C×lX]

for any C-monomial G-set (X,l). Now we show that F is a ring homomorphism. Let (X1,l1) and(X2,l2) beC-monomialG-sets.Then

 F (X1, l1) + (X1, l1)  = F (X1, l1) (X1, l1)  = F (X1 X2, l1 l2)  = C×l1l2(X1 X2) = (X1, l1) (X2, l2) = [C×l1X1] + [C×l2X2].

FormultiplicativityofF we define amap f : C×l1×l2(X1× X2)→ (C ×l1X1)×C(C×l2X2) such that fc,(x1,x2)  = (c,x1)×C (1,x2). Let (k,g) ∈ C × G and  c,(x1,x2)  ∈ C×l1×l2(X1× X2).Then (k, g)fc, (x1, x2)  = (k, g)(c, x1)×C(1, x2)  =(k, g)(c, x1)×C(1, g)(1, x2)  =kcl1(g, x1, gx1), gx1  ×C  l2(g, x2, gx2), gx2  = (kcl1(g, x1, gx1)l2(g, x2, gx2), gx1)×C(1, gx2) = fkcl1(g, x1, gx1)l2(g, x2, gx2), g(x1, x2)  = f(k, g)(c, (x1, x2))  .

Sof isa(C× G)-mapandobviously, f isa(C× G)-isomorphism.Usingf weget  F[X1, l1]· [X2, l2]  = F[X1× X2, l1× l2]  = C×l2×l2(X1× X2) = (C×l1X1)×C(C×l2X2) .

Thus,thedesiredresultfollows. 2

Remark 4.Let (X,l) be a C-monomial G-set. For all x ∈ X, we get a character lx :

Gx→ C defined bylx(g)= l(g,x,x) for g ∈ Gx. On theother hand givena subgroup

U of G anda grouphomomorphism μ : U → C we get aC-monomial G-set (G/U,μ) whereand μ : G/U → •C isthefunctorsuchthatgivengU,kU∈ G/U ifhgU = kU for

someg ∈ G then μ(h,gU,kU )= μ(k−1hg). Moreover, [U,μ]G and [G/U,μ] represents

thesameelement inBC(G).

2.3. The Lefschetzinvariantattachedtoamonomial G-poset

A G-poset X is a partially ordered set (X,≤) with a compatible G-action (that is

gx≤ gy wheneverg∈ G andx≤ y in X).A mapofG-posetsisaG-equivariantmapof posets.Wedenote byG-poset thecategoryoffinite G-posetsobtainedinthisway.

ThereisanobviousfunctorιG: G-set→ G-poset sendingeachfiniteG-settotheset

X orderedbytheequalityrelation, andeachG-equivariantmap toitself.

TheLefschetzinvariantattachedtoafiniteG-poset,whichisanelementofthe Burn-sideringof G hasbeenintroducedin[7] byThévenaz.WewilldefinesimilarlyaLefschetz invariantattachedtoaC-monomialG-posetasanelementoftheC-monomialBurnside ringofG.

2.3.1. The categoryof C-monomialG-posets

Given aG-posetX,weconsiderthecategoryX whose objectsaretheelementsofX

and givenx,y inX theset ofmorphismsfromx toy is

Hom_X(x, y) ={g ∈ G | gx ≤ y}. Now wedefineaC-monomialG-posetasfollows.

Definition 5.A C-monomial G-poset is a pair (X,l) consisting of a G-poset X and a functor l: X → •C.

In other words, foreach x,y ∈ X andg ∈ G such thatgx≤ y, wehave anelement l(g,x,y) ofC,withthepropertythatl(h,y,z)l(g,x,y)= l(hg,x,z) ifh∈ G andhy≤ z,

and l(1,x,x)= 1 foranyx∈ X.

Let(X,l) and(Y,m) beC-monomialG-posets.AmapofC-monomialG-posets from

(X,l) to(Y,m) is apair(f,λ): (X,l)→ (Y,m),wheref : X → Y isamap ofG-posets

and λ: l→ m◦ f is anaturaltransformation.Wedenote the category of C-monomial G-posetsbyCM G-poset.ProductanddisjointunionofC-monomialG-posetsaredefined

as for C-monomial G-sets. When C is the trivial group, we will identify the category

CM G-poset withG-poset.

Remark 6.If (X,l) is a C-monomial G-poset, then for any x∈ X we get a character lx: Gx→ C definedbylx(g)= l(g,x,x).Moreover,ifx≤ y,then

resGx

Gx∩Gylx= res

Gy

Gx∩Gyly

becausewehavethefollowingcommutativediagram:

l(x) l(1,x,y) l(g,x,x) l(y) l(g,y,y) l(x) l(1,x,y) l(y).

LetH be asubgroup ofG and(X,l) beaC-monomialH-set.WeletG×HX tobe

the quotient of G× X by the action of H. The set G×HX isa G-set via theaction

g(u,Hx)= (gu,Hx),forany g∈ G, and(u,Hx)∈ G×HX.Wedefine anorderrelation

≤ onG×HX as

∀(u,Hx), (v,Hy)∈ G ×HX, (u,Hx)≤ (v,Hy)⇔ ∃h ∈ H, u = vh, x ≤ h−1y.

G×HX =



g×HX,

it’senoughtoconsider thechainsoftype(u,Hx0)< ...< (u,Hxn) in G×HX forsome

u∈ G andachainx0< ...< xn inX forsomen∈ N.

Let(u,Hx),(u,Hy) ∈ G×HX and g ∈ G suchthat g(u,Hx) ≤ (u,Hy).Then there

existsh∈ H suchthatgu= uh andhx≤ y.WedefinetheinducedC-monomialG-poset

IndG_H(X,l) of(X,l) asthepair(G×HX,G×Hl) whereG×Hl:G×HX → •Cisdefined

by (G×Hl)  g, (u,Hx), (u,Hy)  = l(h, x, y). Nowshowthat(G×HX,G×Hl) isaC-monomialG-poset.

Let(u,Hx),(u,Hy),(u,Hz)∈ G×HX suchthat

g(u,Hx)≤ (u,Hy)

and

g(u,Hy)≤ (u,Hz)

forsomeg,g∈ G.Then thereexistsomeh,h ∈ H suchthat

gu = uh, gu = uh, hx≤ y, hy≤ z.

Thent= hh∈ H.Moreoverggu= uhh= ut andtx= hhx≤ z.Nowwe get (G×Hl)  gg, (u,Hx), (u,Hz)  = l(t, x, z) = l(hh, x, z) = l(h, y, z)l(h, x, y) = (G×Hl)  g, (u,Hx), (u,Hy)  (G×Hl) 

g, (u,Hy), (u,Hz)

 . Wealsohave(G×Hl)  1,(u,Hx),(u,Hx) 

= 1 forany (u,Hx)∈ G×HX.Thus G×Hl

isafunctor. So IndG_H(X,l) isaC-monomialG-poset.

GivenaC-monomialG-poset (Y,m),therestriction ResG_H(Y,m) of(Y,m) is thepair (ResG_HY,resG

Hm) where Res G

HY istherestrictionoftheG-posetY toH-posetand resGHm

istherestrictionofthefunctorm fromY to ResG_HY .

Proposition7.LetG be afinitegroup.

1. If Y is a finite G-poset, denote by 1Y : Y → •C the trivial functor defined by

1Y(g,x,y)= 1 for any g ∈ G and x,y ∈ Y such that gx≤ y.Then theassignment

Y → (Y,1Y) isafunctorτG from G-poset to CM G-poset.

2. LetH be asubgroup of G.The assignment (X,l)→ IndGH(X,l) is afunctor IndGH : CM H-poset→CM G-poset, and theassignment (Y,m) → ResGH(Y,m) is a functor

3. Moreoverthediagrams H-poset Ind G H τH G-poset τG CM H-poset IndGH CM G-poset

and G-poset Res

G H τG H-poset τH CM G-poset ResGH CM H-poset

of categoriesandfunctorsare commutative.

Proof. 1. Letf : X→ Y beamap ofG-posets.Weset

τG(f ) = (f, 1f) : (X, 1X)→ (Y, 1Y),

where 1f : 1X → 1Y ◦ f is definedby1f x = 1 foranyx∈ X.Obviously(f,1f) isa

mapofC-monomialG-posets andτG is afunctor.

2. Let(f,λ): (X,l)→ (Y,m) beamap ofC-monomialH-posets.Weset thepair IndG_H(f, λ) = (G×Hf, G×Hλ) : (G×HX, G×Hl)→ (G ×HY, G×Hm) where G×Hf : G×HX → G ×HY isdefinedby(G×Hf )(u,Hx)=  u,Hf (x)  and G×Hλ : G×Hl→ (G ×Hm)◦ (G ×Hf )

isdefinedby(G×Hλ)(u,Hx) = λxforany(u,Hx)∈ G×HX.It’sclearthatG×Hf isa

mapofC-monomialG-posets.NowweshowthatG×Hλ isanaturaltransformation.

Let (u,Hx),(u,Hy)∈ G×HX such that g(u,Hx) ≤ (u,Hy) for someg ∈ G. Then

gu= uh andhx≤ y forsomeh∈ H.Sinceλ: l→ m◦ f isanaturaltransformation, weget (G×Hm)  g,u,Hf (x)  ,u,Hf (y)  (G×Hλ)(u,Hx) = m  h, f (x), f (y)λx = λyl(h, x, y) = (G×Hλ)(u,Hy)(G×Hl)  g, (u,Hx), (u,Hy)  .

Nowconsider(idX,idl): (X,l)→ (X,l) where idX : X→ X istheidentity mapon

theH-setX and idl: l→ l◦ idX istheidentitytransformation.Thenweget

IndG_H( idX, idl) = ( idG×HX, idG×Hl).

Now let (f,λ) : (X,l) → (Y,m) and (t,β) : (Y,m) → (Z,r) be the maps of

(G×Ht)◦ (G ×Hf ) = G×H(t◦ f)

and

(G×Hβ)◦ (G ×Hλ) = G×H(β◦ λ).

Thus,

IndG_H(t, β)◦ IndG_H(f, λ) = IndG_H(t, β)◦ (f, λ).

So IndG_H:CM H-poset→CM G-poset isafunctor.

Nowlet(f,λ): (X,l)→ (Y,m) beamapofC-monomialG-posets.Wesetthepair ResG_H(f, λ) = (f|H, λ|H) : ( ResHGX, resGHl)→ ( ResGHY, resGHm)

where f|H: ResGHX → Res G

HY isdefinedastherestrictionofmapofG-posets f to

map of H-posetsand λ|H : resGHl→ resGHm◦ f|H is definedas therestrictionof λ.

Clearly,wegetthat ResG_H :CM G-poset→CM H-poset isafunctor.

3. LetX beanH-poset.Commutativity ofthefirstdiagram followsfrom

τG◦ IndGH(X) = τG(G×HX) = (G×HX, 1G×HX) = Ind

G

H(X, 1X) = IndGH◦ τH(X).

NowletY be aG-poset.Commutativityoftheseconddiagramfollows from

τH◦ ResGH(Y ) = τH( ResGHY ) = ( ResGHY, 1ResG HY)

= ( ResG_HY, resG_H1Y) = ResHG(Y, 1Y) = ResGH◦ τH(Y ). 2

Proposition8.LetG beafinitegroupandH besubgroupofG.Thenthefunctor IndG_H :

CM H-poset→CM G-poset isleftadjointtothefunctor (Y,m)→ ResGH(Y,m).

Proof. WeprovethatforanyC-monomialH-poset(X,l) andanyC-monomialG-poset

(Y,m) wehaveabijection HomCM G



IndG_H(X, l), (Y, m) ∼= HomCM H



(X, l), ResG_H(Y, m) naturalin(X,l) and(Y,m).

Wedefine

ϕ : HomCM G



IndG_H(X, l), (Y, m)→ HomCM H



(X, l), ResG_H(Y, m) where

suchthat

ϕ(f ) : X → ResG_H(Y ) definedbyϕ(f )(x)= f (1,Hx) and

ϕ(λ) : l→ resm ◦ ϕ(f)

defined by ϕ(λ)x = λ(1,Hx) for any x∈ X. Obviously, ϕ(f ) is amap of H-posets. We

need toshowthat

ϕ(λ) : l→ resm ◦ ϕ(f)

is anaturaltransformation.Letx,y∈ X suchthatgx≤ y forsomeg∈ G. Then mh,ϕ(f )(x), ϕ(f )(y)ϕ(λ)x= m  h, f (1,Hx), f (1,Hy)  λ(1,Hx) = λ(1,Hy)l(h, x, y) = ϕ(λ)y(G×Hl)  h, (1,Hx), (1,Hy)  .

Wedefine aninversemap toϕ as

θ : HomCM H



(X, l), ResG_H(Y, m)→ HomCM G

 IndG_H(X, l), (Y, m) where θ : (ψ, β)→θ(ψ), θ(β) suchthat θ(ψ) : G×HX→ Y

definedas θ(ψ)(u,Hx)= uψ(x) and

θ(β) : G×Hl→ m ◦ θ(ψ)

definedas

θ(β)(u,Hx)= m



u, ψ(x), uψ(x)βx

for any (u,Hx) ∈ G×HX.Obviously, themap θ(ψ) is amap ofG-posets.We needto

show that θ(β) is a naturaltransformation. Let (u,Hx),(u,Hy) ∈ G×H X such that

g(u,Hx)≤ (u,Hy) forsomeg∈ G.Thenthereexistssomeh∈ H suchthatgu= uh and

mg, θ(ψ)(u,Hx), θ(ψ)(u,Hy)



θ(β)(u,Hx) = m



g, uψ(x), uψ(y)mu, ψ(x), uψ(x)βx

= mu, ψ(y), uψ(y)mh, ψ(x), ψ(y)βx= m

 u, ψ(y), uψ(y)βyl(h, x, y) = θ(β)(u,Hy)(G×Hl)  h, (uHx), (u,Hy)  .

Clearly,ϕ andθ aremutual inversemaps, andnaturalin(X,l) and(Y,m). 2

2.3.2. The LefschetzinvariantattachedtoaC-monomialG-poset

Let (X,l) be a C-monomial G-poset.The Lefschetz invariant Λ(X,l) of (X,l) is the

elementofBC(G) definedby Λ(X,l)=  x0<...<xn∈GX (−1)n Gx0,...,xn, Res Gx0 Gx0,...,xn(lx0) G

wherex0< ...< xnrunsoverG-representativesofthechainsinX.ThegroupGx0,...,xnis

thestabilizerof theset {x0,...,xn},thatisGx0,...,xn =∩

n

i=0Gxi.Here Res

Gx0

Gx0,...,xn(lx0)

denotes the restriction of the character lx0 introduced in Remark 4. Observe that if

x0< ...< xn isachaininX forsomen∈ N, byRemark6wehave

ResGx0

Gx0,...,xnlx0= Res G_xi Gx0,...,xnlxi

forany0≤ i≤ n.

Let (X,l) be a C-monomial G-poset. Given n ∈ N, let Sdn(X) denote the set of

chainsinX withordern+ 1.Obviously,theset Sdn(X) isaG-set.Then(Sdn(X),ln)

isaC-monomialG-set whereln: Sdn(X)→ •C isthefunctordefinedby

ln(g, x0< ... < xn, y0< ... < yn) = l(g, x0, y0)

foranyx0< ...< xn,andy0< ...< yn in Sdn(X) suchthat

g(x0< ... < xn) = y0< ... < yn

forsomeg∈ G.

Remark9.Given a C-monomial G-poset (X,l), we have the following isomorphism of monomialG-sets:  Sdn(X), ln ∼=



Proof. Let[G/Sdn(X)] be asetof representative oftheG-actionon Sdn(X).Let x=

x0 < ... < xn be a chain in Sdn(X) then there exist some gx ∈ G and a unique

(f, λ) :Sdn(X), ln  →



where f (x)= gxGσx∈ G/Gσx andλx= l(gx−1,gxσx0,σx0).Obviously,

f : Sdn(X)→



G/Gx0,...,xn

is anisomorphismofG-sets.Weshow that

λ : ln→



 ResGx0

G_x0,...,xn(lx0)◦ f

is anaturaltransformation.Letx= x0 < ...< xn, and y = y0< ...< yn be sequences

in Sdn(X) suchthatgx= y forsomeg∈ G.Thereexistauniqueσx,σy∈ [G/Sdn(X)]

such that x = gxσx and y = gyσy for some gx and gy in G. Then x0 = gxσx0 and

y0= gyσy0 soy0= gx0= ggxσx0.Thus,byuniquenessσx0 = σy0 andsog−1y ggx∈ Gσx0.

Then settingr = ResGx0

Gx (lx0)



g,f (x),f (y)λx,wehavethat

r = lx0(g, gxGσx, gyGσy)l(g −1 x , gxσx0, σx0) = lx0(g −1 y ggx)l(gx−1, gxσx0, σx0) = l(g_y−1ggx, x0, x0)l(gx−1, gxσx0, σx0) = l(gx, σx0, gxσx0)l(g, gxσx0, ggxσx0)l(g−1y , ggxσx0, σx0)l(gx−1, gxσx0, σx0) = l(g, x0, y0)l(g−1y , gyσy0, σy0) = ln(g, x, y)λy. 2

ByRemark9,theLefschetzinvariantofaC-monomialG-set(X,l) canbewritten as Λ(X,l)=  x0<...<xn∈GX (−1)n Gx0,...xn, Res Gx0 Gx0,...,xn(lx0) G=  n∈N (−1)nSdn(X), ln  .

It followsthatΛX = ΛτG(X),where ΛX theLefschetzinvariantoftheG-posetX

intro-duced in [2].

Wedefine similarlythereduced Lefschetzinvariant of(X,l)

Λ(X,l)= Λ(X,l)− [G, 1G]G

where 1G is thetrivialcharacterofG.

Lemma 10.LetG be afinite groupandC bean abelian group.

1. Let(X,l) bea C-monomialG-set,viewed as a C-monomial G-posetordered by the equalityrelationonX. ThenΛ(X,l)= [C×lX] inBC(G).

2. Let (X,l) and(Y,m) be C-monomialG-posets.Then Λ(XY,r) = Λ(X,l)+ Λ(Y,m) in

BC(G).

3. Given C-monomialG-posets (X,l) and(Y,m), we have Λ(X×Y,l×m) = Λ(X,l)Λ(Y,m)

in BC(G).

Proof. 1.and2.areclear.

3.Inthefollowingproof usingtheinclusion

BC(G) → Q ⊗ZBC(G)

we identify the elements of BC(G) with their image in Q⊗ZBC(G). We start with

rearrangingthechainsinX×Y asintheproofofLemma 11.2.9in [2].Letn∈ N.Given achain z = z0 < ... < zn inX × Y projection of z on X is denotedby zX and on Y

is denotedby zY. Then zX is achain inX with order i+ 1 for somei ≤ n and zY is

achaininY withorder j + 1 for somej ≤ n such thati+ j = n. Lets_i be thechain

s0< ...< si andtj bethechaint0< ...< tj.Now

Λ(X×Y,l×m)=  n∈N, z∈GSdn(X×Y ) (−1)n Gz, Res Gz0 Gz (lz0) G =  n∈N, z∈ Sdn(X×Y ) (−1)n|Gz| |G|[Gz, Res G_z0 Gz (lz0)]G =  i,j∈N si∈X tj∈Y Γsi,tj where Γsi,tj =  n∈N z∈ Sdn(X×Y ):zX=si, zY=tj (−1)n|Gsi∩ Gtj| |G| Gsi∩ Gtj, Res G_s0 G_si(ls0) Res G_t0 G_tj(mt0) G = |Gsi∩ Gtj| |G| Gsi∩ Gtj, Res Gs0 G_si(ls0) Res Gt0 G_tj(mt0) G  n∈N z∈ Sdn(X×Y ):zX=si, zY=tj (−1)n = |Gsi∩ Gtj| |G| Gsi∩ Gtj, Res Gs0 Gsi(ls0) Res Gt0 Gtj(mt0) G(−1) i+j_. Now, Λ(X×Y,l×m)=  i,j∈N si∈X tj∈Y (−1)i+j|Gsi∩ Gtj| |G| Gsi∩ Gtj, Res G_s0 G_si(ls0) Res G_t0 G_tj(mt0) G.

Ontheother hand Λ(X,l)Λ(Y,m)=  i∈N si∈GX (−1) Gsi, Res G_s0 Gsi(ls0)  j∈N tj∈GY (−1)j Gtj, Res G_t0 Gtj(mt0) G =  i,j∈N si∈X tj∈Y GigGtj⊆G (−1)i+j|Gsi||Gtj| |G|2 Gsi∩ g_G tj, Res G_s0 G_si(ls0) Res g_G t0 g_G tj( g_m t0) G =  i,j∈N si∈X tj∈Y g∈G (−1)i+j |Gsi||Gtj| |G|2_|G sigGtj| Gsi∩ g_G tj, Res G_s0 Gsi(ls0) Res g_G t0 g_G tj( g_m t0) G =  i,j∈N si∈X tj∈Y g∈G (−1)i+j|Gsi∩ g_G tj| |G|2 Gsi∩ g_G tj, Res G_s0 G_si(ls0) Res g_G t0 g_G tj( g_m t0) G =  i,j∈N si∈X tj∈Y g∈G (−1)i+j|Gsi∩ Ggtj| |G|2 Gsi∩ Ggtj, Res Gs0 G_si(ls0) Res Ggt0 G_gtj(mgt0) G =  i,j∈N si∈X tj∈Y (−1)i+j|Gsi∩ Gtj| |G| Gsi∩ Gtj, Res G_s0 Gsi(ls0) Res G_t0 Gtj(mt0) G.

Thus, Λ(X×Y,l×m)= Λ(X,l)Λ(Y,m). 2

The first assertionof Lemma 10tells us thatevery positiveelement of BC(G) is in

of the form Λ(X,l) for some C-monomial G-poset (X,l). Now consider the poset X =

{a,b,c,d,e} with theordering{a≤ c,a≤ d,a≤ e,b≤ c,b≤ d,b≤ e}.Considertrivial

G-actionon X.Then ΛτG(X)=−1BC(G). SoasaconsequenceofLemma10weget the

following corollary.

Corollary 11. Any element of themonomial Burnside ring can be expressed as the Lef-schetz invariantofsome (nonunique)monomial G-poset.

Proposition12. LetH beasubgroupof G.GivenaC-monomialH-poset(X,l),wehave

IndG_H(Λ(X,l)) = ΛIndG H(X,l).

Proof. Since IndG_H(Λ(X,l)) =  n∈N (−1)nIndG_HSdn(X), ln  ,

weneedtoshowthatthereexists aC-monomialG-setisomorphismbetween  G×H Sdn(X), G×Hln  and  Sdn(G×HX), (G×Hl)n  foranyn∈ N. Wedefine (fn, id) :  G×H Sdn(X), G×Hln  →Sdn(G×HX), (G×Hl)n  where fn : G×H Sdn(X)→ Sdn(G×HX) suchthat fn(u,Hx0< ... < xn) =  (u,Hx0) < ... < (u,Hxn)  foranychain(u,Hx0< ...< xn) inG×H Sdn(X).

Let(u0,Hx0)< ...< (un,Hxn) beachainin Sdn(G×HX).There existsomehi∈ H

suchthatuihi = ui+1 andhi−1xi< xi+1 forall0≤ i≤ n− 1.Then

fn



u0,Hx0< h0x1< ... < h0...hn−1xn



= (u0,Hx0) < ... < (un,Hxn).

Obviously,fn isamap ofG-setsandinjective.

Now, we show thatG×Hln = (G×Hl)n◦ fn. We consider an element k∈ G, and

chains(u,Hx0< ...< xn) inG×H Sdn(X) suchthat

k(u,Hx0< ... < xn) = (v,Hy0< ... < yn).

Thereexists someh∈ H suchthatku= vh and hxi= yi forall0≤ i≤ n.Then

(G×Hl)n  k, fn(u,Hx0< ... < xn), fn(v,Hy0< ... < yn)  = (G×Hl)n  k, (u,Hx0) < ... < (u,Hxn), (v,Hy0) < ... < (v,Hyn)  = ln(h, x0< ... < xn, y0< ... < yn) = (G×Hln)  k, (u,Hx0< ... < xn), (v,Hy0< ... < yn)  . 2

Let (X,l) beaG-poset andletx∈ X.Then thepairs(]x,·[X,l>x) and (]·,x[X,l<x)

are C-monomialGx-posetswhere

]x,·[X={y ∈ X | x < y}, ]·, x[X={y ∈ X | y < x}

whichareGx-posetsandl>x: ]x,·[X→ •C andl<x: ]·, x[X → •C aretherestrictionsof

thefunctorl.

Lemma 13.Let(X,l) be amonomial G-poset. Wehave

Λ(X,l)=−  x∈[G/X] IndG_Gx[Gx, lx]Gx· Λ]x,·[X  . Proof. Λ(X,l)=  x0<...<xn∈GX (−1)n Gx0,...,xn, Res Gx0 Gx0,...,xn(lx0) G =  x0∈GX  x1<...<xn∈GX:x0<x1 (−1)n Gx0,...,xn, Res G_x0 Gx0,...,xn(lx0) G =  x0∈GX IndG_Gx0  x1<...<xn∈_Gx0]x0,·[X (−1)n Gx0,...,xn, Res G_x0 G_x0,...,xn(lx0) G_x0 =  x0∈GX IndG_Gx0 Gx0, lx0 G_x0  x1<...<xn∈_{Gx0 ]x0,·[X} (−1)n Gx0,...,xn, 1Gx0,..,xn G_x0 =−  x∈GX IndG_Gx[Gx, lx]Gx· Λ]x,·[X  . 2

Remark 14.Wecandefine theopposite ofaC-monomial G-poset (X,l) as follows. We consider the pair (Xop_,lop_{) where X}op _{is the opposite G-poset with the order ≤}op

definedby

∀x, y ∈ X, g ∈ G, gx ≤op_{y⇔ y ≤ gx}

and lop: Xop_{→ •}

C isdefinedby

lop(g, x, y) = l−1(g−1, y, x)

for any x,y ∈ Xop and g ∈ G such that gx ≤op y. Obviously, the pair (Xop,lop) is a C-monomial G-poset. Moreover the assignment (X,l) → (Xop,lop) is a functor

CM G-poset→CM G-poset:if(f,λ): (X,l)→ (Y,m) isamapofC-monomialG-posets,

thenf : Xop_{→ Y} op_{isamapofG-posetsandforanygx≤}op_x_{,wegetthecommutative}

l(x) λx lop(g,x,x) m◦ f(x) mopg,f (x),f (x) l(x) λx m◦ f(x). Observethat(lop₎

x(g)= l−1(g−1,x,x)= l(g,x,x)= lx(g),foranyx∈ X andg∈ Gx.

ItfollowsthatΛ(X,l)= Λ(Xop_,lop₎.

Let(f,λ): (X,l)→ (Y,m) be amap ofC-monomialG-posets.Given y ∈ Y , follow-ing [3] we set

fy ={x ∈ X | f(x) ≤ y}, fy={x ∈ X | f(x) ≥ y}

which are both Gy-posets. We denote by (fy,l|fy) the C-monomial G_y-poset where

l_|fy : fy → •_C istherestrictionofthe functorl. Similarly,wedenote by(f_y,l_|fy) tobe

C-monomialGy-posetwhere l|fy : fy→ •C istherestrictionofthefunctor l.

Example15.Let(f,λ): (X,l)→ (Y,m) beamap ofC-monomialG-posets.Wedefinea

G-posetX∗f,λY withunderlyingG-setX Y asfollows:forz,z∈ X  Y ,weset

z≤ z⇔ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ z, z∈ X and z≤ z ∈ X z, z∈ Y and z≤ z ∈ Y z∈ X, z∈ Y and f(z) ≤ z∈ Y .

Wedefinethefunctorl∗f,λm: X Y → •C by

(l∗f,λm)(g, z, z) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ l(g, z, z) if z, z∈ X m(g, z, z) if z, z∈ Y m(g, f (z), z)λz if z∈ X, z∈ Y ,

foranyz,z∈ X ∗f,λY andg∈ G suchthatgz≤ z.

Nowletz1,z2,z3∈ X ∗f,λY andg,g∈ G suchthatgz1≤ z2 andgz2≤ z3.Weaim

toshowthat

(l∗f,λm)(gg, z1, z3) = (l∗f,λm)(g, z2, z3)(l∗f,λm)(g, z1, z2).

Wehavefour casesto consider: • z1,z2,z3∈ X,

• z1∈ X andz2, z3∈ Y ,

• z1,z2,z3∈ Y .

Inthefirstcaseweget

(l∗f,λm)(gg, z1, z3) = l(gg, z1, z3) = l(g, z2, z3)l(g, z1, z2)

= (l∗f,λm)(g, z2, z3)(l∗f,λm)(g, z1, z2).

Inthesecond case,using thenaturalityofλ weget

(l∗f,λm)(gg, z1, z3) = m  gg, f (z1), z3  λz1 = m  g, f (z2), z3  mg, f (z1), f (z2)  λz1 = l(g, z1, z2)m  g, f (z2), z3  λz2 = (l∗f,λm)(g _{, z} 2, z3)(l∗f,λm)(g, z1, z2).

Inthethirdcase,weget (l∗f,λm)(gg, z1, z3) = m  gg, f (z1), z3  λz1 = m  g, f (z2), z3  mg, f (z1), f (z2)  λz1 = (l∗f,λm)(g, z1, z2)(l∗f,λm)(g, z2, z3).

Inthefourthcase

(l∗f,λm)(gg, z1, z3) = m  gg, z1, z3  = m(g, z2, z3)m(g, z1, z2) = (l∗f,λm)(g, z2, z3)(l∗f,λm)(g, z1, z2).

Letz∈ X ∗f,λY thenobviously wehave(l∗f,λm)(1,z,z)= 1.Thus,(X∗f,λY,l∗f,λm)

is aC-monomialG-poset.

Lemma 16. Let (f,λ) : (X,l) → (Y,m) be a map of C-monomial G-posets. Then

Λ(X∗f,λY,l∗f,λm)= Λ(Y,m).

Proof. 1. Letz∈ Z = X ∗f,λY .Ifz∈ X considerthemapg : ]z,·[Z→ [f(z),·[Y defined

by

g(t) =



f (t) if t∈ X

t if t∈ Y .

Letg: f (z),· →z,· definedbyg(s)= s.Then g andg aremapsofGz-posets

such that g◦ g = Id and Id ≤ g ◦ g. So if z ∈ X using [[3], Lemma 4.2.4 and Proposition 4.2.5],weget Λ]z,·[= Λ[f (z),·[= 0.Thus,

Λ(X∗f,λY,l∗f,λm)=−  z∈[G\X∗f,λY ] IndGGz([Gz, lz]Gz· Λ]z,·[) =−  y∈[G\Y ]

IndG_Gy([Gy, ly]Gy · Λ]y,·[) = Λ(Y,m). 2

Asaconsequence,wegiveananalogueofProposition 4.2.7 in [3], whichinturn was inspiredbyamuchdeepertheoremof Quillenin [6].

Proposition17. Let(f,λ): (X,l)→ (Y,m) be amapof C-monomialG-posets.Then in BC(G) Λ(Y,m)= Λ(X,l)+  y∈G\Y IndG_Gy( Λfy Λ_(]y,·[ Y,m>y)). Λ(Y,m)= Λ(X,l)+  y∈G\Y

IndG_Gy( Λfy Λ(]·,y[Y,m<y)).

Proof. We follow the proof of Proposition 4.2.7 in [3]. For any n ∈ N, any chain z = z0< ...< zn∈ Sdn(X∗f,λY ) canbeoftwotypes,dependingonzn∈ X orzn ∈ Y .For

asequence z ofthefirsttypeweget

ResGzn

G_z0,...,zn(l∗f,λm)nzn= Res

Gzn

G_z0,...,znln.

Nowasequence z of thesecond typehas asmallest element y = zi inY , thus, wecan

writethesequenceas

x0< ... < xi−1< y < y0< ... < yn−i−1

suchthatx0< ...< xi−1 isin Sdi−1(fy),and y0< ...< yn−i−1 isin Sdn−i−1(]y,·[Y).

Weget

ResGzn

G_z0,...,zn(l∗f,λm)nzn= Res

Gy

G_z0,...,zn(m).

Letx_i−1denotethechainx0< ...< xi−1andy_n−i−1denotethechainy0< ...< yn−i−1.

Then,byLemma10andLemma16weget Λ(Y,m)= Λ(X∗f,λY,l∗f,λm)=  n∈N (−1)nSdn(X∗f,λY ), (l∗f,λm)n  =  n∈N z0<...<zn∈ Sdn(l∗f,λm) (−1)n Gz0,...,zn, Res Gzn G_z0,...,zn(l∗f,λm)n G = n∈N (−1)nSdn(X), ln 

+  y∈[G\Y ] IndG_Gy n  i=0  x_i−1∈ Sdi−1(fy) y_n−i−1∈ Sdn−i−1(]y,·[Gy)

Gxi−1,y,yn−i−1, Res

Gy G_{xi−1,y,yn−i−1}my Gy = Λ(X,l)+  y∈[G\Y ] IndG_Gy Λ(fy_,1 f y) Λ(]y,·[Y,m>y)  .

Forthesecond assertionwe considertheoppositemap (f, λ) : (Xop, lop)→ (Yop, mop)

Since wehaveΛ(X,l)= Λ(Xop_,lop₎ byRemark14,theresultfollows. 2

Corollary 18.Let(f,λ): (X,l)→ (Y,m) beamapof C-monomialG-posets.If Λfy = 0

forally∈ Y (resp.if Λfy = 0 forally∈ Y ),then ΛX,l= ΛY,m.

Remark 19.The assumption of this corollary is fulfilled in particular if f : X → Y

admitsarightadjointg,inotherwordsifthere existsamap ofposetsg : Y → X such

thatf (x)≤ y ⇔ x≤ g(y) foranyx∈ X andy∈ Y , i.e.equivalentlyiff◦ g(y)≤ y and g◦ f(x)≤ x foranyx∈ X andanyy∈ Y .

Now we set some notation. Given a C-monomial G-set (X,l), we can rewrite its Lefschetz invariantas Λ(X,l)=  x0<...<xn∈GX (−1)n Gx0,...xn, Res Gx0 G_x0,...,xn(lx0) G =  (V,ν)∈Gch(G) γ_V,νX,l[V, ν]G where γ_V,νX,l=  x0<...<xn∈GX (G_x0,...,xn, ResGx0_Gx0,...,xnl_x0)=G(V,ν) (−1)n = 1 |NG(V, ν) : V|  x0<...<xn∈X (G_x0,...,xn, ResGx0_Gx0,...,xnl_x0)=(V,ν) (−1)n.

Given aC-monomialG-poset (X,l) welettheset(X,l)U,μ tobe

(X, l)U,μ={x ∈ XU | ResGx

U lx= μ}

where (U,μ) is a subcharacter of G. Then given a C-subcharacter (U,μ) ∈ ch(G) we

χ(X, l)U,μ=  n∈N x0<...<xn∈XU ResGx0_Gx0,...,xnl_x0=μ (−1)n=  (V,ν)∈ ch(G) U⊆V ResV Uν=μ mX,l_V,ν where mX,l_V,ν =  n∈N x0<...<xn∈X (Gx0,...,xn, ResGx0_Gx0,...,xnlx0)=(V,ν) (−1)n. Now|NG(V,ν): V|mX,lV,ν= γ X,l

V,ν.Usingthis factweprovethefollowinglemma.

Lemma 20. Let(X,l) and (Y,m) be C-monomial G-posets then Λ(X,l) = Λ(Y,m) if and

onlyif χ(X,l)U,μ= χ(Y,m)U,μ forevery C-subcharacter (U,μ) of G.

Proof. AssumeΛ(X,l)= Λ(Y,m).Then

 (V,ν)∈Gch(G) γ_V,νX,l[V, ν]G =  (V,ν)∈Gch(G) γ_V,νY,m[V, ν]G  (V,ν)∈Gch(G) (γ_V,νX,l− γ_V,νY,m)[V, ν]G= 0.

Soγ_V,νX,l = γ_V,νY,mandthen m_V,nuX,l = mY,m_V,nuforeveryC-subcharacter(V,ν) ofG.Weget 

(U,μ)≤(V,ν)∈Gch(G)

mX,l_V,ν = 

(U,μ)≤(V,ν)∈Gch(G)

mY,m_V,ν.

Thus,χ(X,l)U,μ= χ(Y,m)U,μforeveryC-subcharacter(U,μ) ofG.

Conversely,assumethatχ(X,l)U,μ= χ(Y,m)U,μforeveryC-subcharacter(U,μ)

ofG.Then  (U,μ)≤(V,ν)∈ ch(G) mX,l_V,ν =  (U,μ)≤(V,ν)∈ ch(G) mY,m_V,ν,  (U,μ)≤(V,ν)∈ ch(G) (mX,l_V,ν− mY,m_V,ν) = 0.

Letz bethematrixwith thecoefficients

z(U, μ; V, ν) =(U, μ) ≤ (V, ν) =



1 if (U, μ)≤ (V, ν) 0 otherwise

forany C-subcharacters (U,μ),(V,ν). Ifwe list theC-subcharacters innon-decreasing order of size of thesubgroups, the matrixz is uppertriangular with nonzero diagonal

coefficients. Thus, z isnonsingular andso mX,l_V,ν = m_V,νY,m. Thisimplies γX,l_V,ν = γ_V,νY,m. We get Λ(X,l)=  (V,ν)∈Gch(G) γ_V,νX,l[V, ν]G=  (V,ν)∈Gch(G) γ_V,νY,m[V, ν]G= Λ(Y,m).

This provesthelemma. 2 3. Generalizedtensorinduction

LetG andH befinitegroups.A setU isa(G,H)-biset ifU isaleftG-setandright

H-set such thattheG-actionand the H-actioncommute.Any (G,H)-biset U is aleft (G× H)-setwith thefollowing action:

∀u ∈ U, (g, h) ∈ G × H (g, h) · u = guh−1_.

AC-monomial(G× H)-set(U,λ) willbecalledaC-monomial(G,H)-biset,andusually denotedbyUλ forsimplicity.

Now letUλ beaC-monomial(G× H)-setandu,u∈ U.Thenthesetofmorphisms

from u tou inU is

Hom_U(u, u) ={(g, h) ∈ G × H | gu = uh}.

If(g,h)∈ Hom_U(u,u),wedenote theimageof(g,h) underλ byλ(g,h,u,u).

LetUλbeaC-monomial(G,H)-bisetandVρbeaC-monomial(H,K)-biset.Consider

theset

Uλ◦ Vρ={(u, v) ∈ U × V | ∀h ∈ Hu∩ Hv, λ(1, h, u, u)ρ(h, 1, v, v) = 1}.

Theset Uλ◦ Vρ isanH-setwiththeaction

∀(u, v) ∈ Uλ◦ Vρ,∀h ∈ H, h(u, v) = (uh−1, hv).

Indeed, theconditionthatweimpose onUλ◦ Vρ amountstosaying thatgiven(u,v)∈

Uλ◦ Vρ, the linear character ξu,v : h → λ(1,h,u,u)ρ(h,1,v,v) of Hu∩ Hv is trivial.

Moreover we haveξux,x−1v(h)= ξu,v(xhx−1)= 1 for x∈ H and h∈ Hux∩ Hx−1v,i.e.

xhx−1 ∈ Hu∩ Hv.

WeletUλ◦HVρdenotethesetofH-orbitsonUλ◦ Vρ and(u,Hv) denotetheH-orbit

containing(u,v).Theset Uλ◦HVρ is(G,K)-bisetwiththeaction

(u,Hv)∈ Uλ◦HVρ, (g, k)∈ G × K, g(u,Hv)k = (gu,Hvk).

WeobtainaC-monomial(G,K)-biset(Uλ◦HVρ,λ×ρ),whereλ×ρ isdefinedasfollows:

thenthere exists h∈ H such thatgu= uh and hv = vk. This element h neednotbe unique,butitiswelldefineduptomultiplicationontherightbyanelementofHu∩ Hv.

Weset

(λ× ρ)g, k, (u,Hv), (u,Hv) = λ(g, h, u, u)ρ(h, k, v, v),

whichdoesnotdependonthechoiceofh,bythedefiningpropertyofUλ◦ Vρ.Notethat

Uλ◦HVρ= U×HV whenV isaleftfree(H,K)-biset,orwhenλ andρ arebothequal

tothetrivialfunctor.

GivenaC-monomialG-poset(X,l),welettU,λ(X,l) bethesetofG-equivariantmaps

f : U→ X suchthat

lg, f (u), f (u)= λ(g, 1, u, u)

forallu∈ U andg∈ Gu.ThentU,λ(X,l) isanH-posetwiththeaction(hf )(u)= f (uh),

foranyh∈ H,foranyf ∈ tU,λ(X,l),foranyu∈ U.Theorder≤ isgivenas follows:

∀f, f_{∈ t}

U,λ(X, l), f ≤ f ⇔ ∀u ∈ U, f(u) ≤ f(u) in X.

NowwedefineafunctorLU,λ:tU,λ(X, l)→ •C.Letf ,f∈ tU,λ(X,l) andh∈ H such

thathf ≤ f. Wechoose a set [G\U] of representatives of G-orbits of U . Then for all

u∈ U thereexistsomegh,u∈ G andauniqueσh(u)∈ [G\U] suchthat

uh = gh,uσh(u).

Sincehf ≤ f,wegetgh,uf



σh(u)



≤ f_{(u),and weset}

LU,λ(h, f, f) =  u∈[G\U] l  gh,u, f  σh(u)  , fuλ−1gh,u, h, σh(u), u  .

Nowweshowthatthisdefinitiondoesnotdependonthechoiceofgh,u.Assumethat

thereexist gh,u,gh,u∈ G suchthat

uh = gh,uσh(u) = gh,u σh(u).

Sothereexistsw∈ Gσh(u) suchthatgh,u= g

 h,uw. Weget l  w, fσh(u)  , fσh(u)  = λw, 1, σh(u), σh(u)  .

Furthermore,wegetthefollowingcommutativediagram:

σh(u) w gh,u σh(u) gh,u u

Thus, LU,λ(h, f, f) =  u∈[G\U] l  gh,u, f  σh(u)  , fuλ−1gh,u, h, σh(u), u  =  u∈[G\U] l  g_h,u w, fσh(u)  , fuλ−1g_h,u w, h, σh(u), u  =  u∈[G\U] l  g_h,u , fσh(u)  , fuλ−1g_h,u, h, σh(u), u  .

Definition 21.The above construction TU,λ : (X,l) →



tU,λ(X,l),LU,λ



is called the

generalized tensorinduction forC-monomialG-posets,associatedto (U,λ).

Lemma 22. LetG and K befinite groupsand U be a(G,K)-biset. Thenthere exists a bijection betweenthesets{(u,t)| u∈ [G\U/K],t∈ [(K ∩ Gu_{)\K]} and [G\U].}

Proof. Let u ∈ [G\U/K] andt ∈ [(K ∩ Gu)\K] then there exist somegt,u ∈ G and a

uniqueσt(u)∈ [G\U] suchthat

ut = gt,uσt(u).

Wedefineψ :{(u,t)| u∈ [G\U/K], t∈ [(K ∩ Gu_{)\K]}→ [G\U] byψ(u,t)= σ}

t(u). 2

Lemma 23. LetG and H befinite groups, (U,λ) be amonomial (G,H)-biset and(X,l)

be aC-monomialG-poset. 1. tU,λ(X,l),LU,λ  is aC-monomialH-poset. 2. tU,λ(X,l),LU,λ 

does not depend on the choice of representative set [G\U], up to isomorphism.

Proof. 1. We show that LU,λ : tU,λ(X, l) → •C is a functor. Let h, h ∈ H and

f,f, f ∈ tU,λ(X,l) such that hf ≤ f and hf ≤ f. Let u ∈ [G\U]. Then

there exist somegh,u,gh,u, ghh,u inG anduniqueelements σh(u), σh(u), σhh(u)

in[G\U] such that

uh = gh,uσh(u), uh= gh,uσh(u), uhh = ghh,uσhh(u).

Alsothereexist somegh,σh(u)∈ G andauniqueσh



σh(u)



∈ [G\U] suchthat

σh(u)h = gh,σh(u)σh  σh(u)  . Nowweget uhh = gh,ugh,σh(u)σh  σh(u) 

and

σhh(u) = σh



σh(u).

Thenthere existsw∈ Gσh h(u) suchthat

ghh,u= gh,ugh,σh(u)w.

Wehavethefollowingcommutativediagram:

σhh(u) w gh h,u σhh(u) gh ,ugh,σ_h(u) uhh

Ontheotherhandsincew∈ Gσh h(u),weget

l  w, fσhh(u)  , fσhh(u)  = λw, 1, σhh(u), σhh(u)  .

Thus,setting L= LU,λ(hh,f,f),wehave

L =  u∈[G\U] l  ghh,u, f  σhh(u)  , fuλ−1ghh,u, hh, σhh(u), u  =  u∈[G\U] l  gh,ugh,σh(u)w, f  σhh(u)  , fuλ−1gh,ugh,σh(u)w, h _{h, σ} hh(u), u  =  u∈[G\U] l  gh,ugh,σh(u), f  σhh(u)  , fuλ−1gh,ugh,σh(u), h _{h, σ} hh(u), u  = L(h, f, f)L(h, f, f).

Moreover,given f ∈ TU,λ(X,l) wehave

L(1, f, f ) = 

u∈[G\U]

l1, f (u), f (u)λ−1(1, 1, u, u) = 1.

Thus,LU,λ:tU,λ(X, l)→ •C isafunctor.

2. Let h∈ H and f ,f ∈ tU,λ(X,l) such thathf ≤ f. LetS = [G\U] and letS be

theanotherchoiceofrepresentatives.Ifu∈ S thenthereexistsomeau∈ G,anda

uniqueu∈ S suchthatu= auu.Then thereexist somegh,auu,gh,u∈ G, a unique

σ_h(auu)∈ S,and auniqueσh(u)∈ S suchthat

auuh = gh,auuσ

 h(auu)

and

uh = gh,uσh(u).

Then

auuh = augh,uσh(u) = augh,ua−1_σh(u)aσh(u)σh(u).

So σ_h(auu)= aσh(u)σh(u). Note that aσh(u)σh(u) ∈ S

_{.We get the following}

com-mutativediagram: aσh(u)f  σh(u) augh,ua−1_σh(u) a−1_σh(u) auf  u fσh(u)  gh,u f (u). au

Thus,settingL= L_U,λ(h,f,f),wehave

L =  auu∈S l  augh,ua−1_σh(u), f  aσh(u)σh(u)  , fauu  

λ−1augh,ua−1_σh(u), h, aσh(u)σh(u), auu

 = L_U,λ(h, f, f) = LU,λ(h, f, f)αfα−1f where αf =  u∈S lau, f(u), auf(u)  λ−1au, 1, u, auu  and α−1_f =  u∈S la−1_u , auf (u), f (u)  λ−1a−1_u , 1, auu, u  . 2

Proposition 24.LetG andH be finitegroupsand(U,λ) beaC-monomial(G,H)-biset.

1. Let(X,l),(X,l) be C-monomialG-posetsthen

TU,λ



(X, l)× (X, l) ∼= TU,λ(X, l)× TU,λ(X, l).

Proof. 1. is clear.

2. Let(ϕ,β): (X,l)→ (Y,m) beamap ofC-monomial G-posets.Wedefineamap of

C-monomialG-posets  TU,λ(ϕ), TU,λ(β)  :tU,λ(X, l), L  →tU,λ(Y, m), M  where

TU,λ(ϕ) : tU,λ(X, l)→ tU,λ(Y, m)

suchthatTU,λ(ϕ)(f )= ϕ◦ f and

TU,λ(β) : L(f )→ M ◦ TU,λ(ϕ)(f ) suchthat TU,λ(β) =  u∈[G\U] βf (u)

for any f ∈ tU,λ(X,l). Clearly, ϕ◦ f : U → X → Y is a map of G-posets. Since

given g ∈ Gu and u∈ U the map β : l→ m◦ ϕ is natural,we have thefollowing

commutativediagram: lf (u) βf (u) l(g,f (u),f (u)) m◦ ϕf (u) m(g,ϕ◦f(u),ϕ◦f(u)) lf (u) βf (u) m◦ ϕf (u). So βf (u)l  g, fu, fu= m 

g, ϕf (u), ϕf (u)βf (u).

Sinceg∈ Gf (u),wehave

l  g, fu, fu= λ(g, 1, u, u). Thenweget m 

g, ϕf (u), ϕf (u)= λ(g, 1, u, u). Thus,ϕ◦ f ∈ tU,λ(Y,m).

Now weshowthat

TU,λ(β) : L→ M ◦ TU,λ(ϕ)

is anatural transformation. Let f,f ∈ tU,λ(X,l) and h ∈ H such that hf ≤ f. We

show thatthefollowingdiagram iscommutative: Lf TU,λ(β)f L(h,f,f) M◦ TU,λ(ϕ)(f ) M(h,ϕ◦f,ϕ◦f) Lf TU,λ(β)f  M◦ TU,λ(ϕ)(f).

Letu∈ [G\U].Thenthereexistsomegh,u∈ G andauniqueσh(u)∈ [G\U] suchthat

uh = gh,uσh(u).

Since β : l→ m◦ ϕ is a naturaltransformation,we obtain the following commutative diagram:

l 

fσh(u)

 β_{f (σh(u))} lgh,u, f (σh(u)), f(u)

 m◦ ϕ  fσh(u)  m  gh,u, ϕ  fσh(u)  , ϕfu lf(u) βf  (u) m◦ ϕf(u).

Usingthecommutativityoftheabovediagram,andsettingT = TU,λ(β)f◦L(h,f,f),

we get T = TU,λ(β)f   u∈[G\U] l  gh,u, f  σh(u)  , fuλ−1gh,u, h, σh(u), u  =  u∈[G\U] βf(u)l  gh,u, f  σh(u) 

, f(u)λ−1gh,u, h, σh(u), u

 =  u∈[G\U] m  gh,u, ϕ  fσh(u)  , ϕfuλ−1gh,u, h, σh(u), u  β_f_σh(u) = M(h, ϕ◦ f, ϕ ◦ f)βf.

SoTU,λ(β): L→ M◦ TU,λ(ϕ) isanaturaltransformation.Thus,

 TU,λ(ϕ), TU,λ(β)  :tU,λ(X, l), L  →tU,λ(Y, m), M  is amapofC-monomialG-posets. 2

Lemma25. LetG,H andK be finite groups.If U isa(G,H)-biset andV is aleft free

(H,K)-biset, then themap (u,v)∈ U × V → (u,Hv)∈ U ×HV restricts to abijection

π : [G\U]× [H\V ] → [G\(U ×H V )], where brackets denote sets of representatives of

orbits.

Proof. For(u,v)∈ U × V ,there existsv0∈ [G\V ] and h∈ H suchthatv = hv0. Then

thereexistsu0∈ [G\U] andg∈ G suchthatuh= gu0.Then(u,Hv)= g(u0,Hv0).Hence

π issurjective.Now if(u0,v0) and (u1,v1) are pairsin[G\U]× [H\V ] which lieinthe

sameG-orbit,there exists g∈ G and h∈ H such that(gu0,v0)= (u1h−1,hv1).Hence

hv1 = v0, so v0 = v1 = hv1, and h = 1 since H act freely on V . Then gu0 = u1, so

u0= u1, andπ isinjective. 2

Proposition26. LetG,H andK befinite groups.

1. Let (•,1) be the C-monomial G-posetwhere • is G-posetwith one element and 1:

•→ •C isthefunctor suchthat1(g,•,•)= 1.ThenTU,λ(•,1)= (•,1).

2. Let(∅,z) betheempty C-monomial(G,H)-poset.ThenT_∅,z istheconstant functor with value (•,1).

3. Let(U,λ) and(U,λ) be C-monomial(G,H)-bisetsandlet(X,l) beaC-monomial G-poset then

TUU_,λλ(X, l) = TU,λ(X, l)TU,λ(X, l).

4. Let idG stand for the identity (G,G)-biset. Then TidG,1(X, l) = (X,l) for any

C-monomialG-poset (X,l).

5. Let (V,ρ) be a C-monomial left free (H,K)-biset, and (U,λ) be a C-monomial

(H,G)-biset.Then

TV,ρ◦ TU,λ= TU×HV,λ×ρ.

Proof. 1.,2.,3.and 4.areclear.

5. Note that since V is left free, we have Uλ ◦H Vρ ∼= GU×HVK. Let (X,l) be a

C-monomialG-poset.Weneedto showthat  tV,ρ  tU,λ(X, l), LU,λ  , LV,ρ◦ LU,λ  =tU×HV,λ×ρ(X, l), LU×HV,λ×ρ  .

WedefineaK-posetmapϕ: tV,ρ



tU,λ(X,l),LU,λ



→ tU×HV,λ×ρ(X,l) suchthat

ϕ(f )(u,Hv) = f (v)(u)

foranyf ∈ tV,ρ



tU,λ(X,l),LU,λ



and(u,Hv)∈ U ×HV .It’s clearthatthemapϕ(f ) is

Letg∈ G(u,Hv).NotethatsinceV isH-free,wehaveg∈ Gu.Then l  g, ϕf(u,Hv), ϕ  f(u,Hv)  = l  g, f (v)(u), f (v)(u)  = λ(g, 1, u, u)ρ(1, 1, v, v) = (λ× ρ)g, 1, (u,Hv), (u,Hv)



.

and soϕ(f )∈ tU×HV,λ×ρ(X,l).

Now wedefineamap

θ : tU×HV,λ×ρ(X, l)→ tV,ρ



tU,λ(X, l), LU,λ



suchthatθ(t)(v)(u)= t(u,Hv) foranyt∈ tU×HV,λ×ρ(X,l),u∈ U and v∈ V .Weshow

that θ(t)∈ tV,ρ



tU,λ(X,l),LU,λ



. Indeed, the map θ(t) isclearly a map of H-sets and moreover,sinceV isH-free,wehaveHv= 1 for anyv∈ V .Then

LU,λ



1, θ(t)(v), θ(t)(v)= 1 = ρ(1, 1, v, v).

Clearly,θ(t)(v) is amapof G-sets.Letg∈ Gu.Then g∈ G(u,Hv),and weget

l 

g, θt(v)(u), θt(v)(u)= lg, t(u,Hv), t(u,Hv)

 = λ(g, 1, u, u)ρ(1, 1, v, v) = λ(g, 1, u, u). Soθ(t)∈ tV,ρ  tU,λ(X,l),LU,λ  .

NowweshowthatLV,ρ◦LU,λ=LU×HV,λ×ρ.Letk∈ K andf,f ∈ tV,ρ



tU,λ(X,l),LU,λ

 such thatkf ≤ f. Letv∈ [H\V ].Then there existauniqueσk(v)∈ [H\V ] and some

hk,v ∈ H suchthat

vk = hk,vσk(v).

Let u∈ [G\U]. Thenthere exist auniqueσhk,v(u)∈ [G\U] and someghk,v,u∈ G such

that

uhk,v= ghk,v,uσhk,v(u).

Then

(u,Hv) = (uhk,vhk,v−1,Hv) = (uhk,v,Hhk,v−1v)

=ghk,v,uσhk,v(u),Hσk(v)k−1  = ghk,v,u  σhk,v(u),Hσk(v)  k−1. Weget (u,Hv)k = ghk,v,u  σhk,v(u),Hσk(v)  .