Group actions on posets

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Journal of Algebra 285 (2005) 439–450

www.elsevier.com/locate/jalgebr

Group actions on posets✩

Eric Babsona, Dmitry N. Kozlovb,∗

a Department of Mathematics, University of Washington, Seattle, WA 98105, USAb Departments of Computer Science and Mathematics, Eidgenössische Technische Hochschule

Zürich, Switzerland

Received 12 May 2000

Available online 22 January 2005

Communicated by Michel Broué

Abstract

In this paper we study quotients of posets by group actions. In order to define the quotient cowe enlarge the considered class of categories from posets to loopfree categories: categoriesnontrivial automorphisms and inverses. We view group actions as certain functors and defiquotients as colimits of these functors. The advantage of this definition over studying the qposet (which in our language is the colimit in the poset category) is that the realization of the qloopfree category is more often homeomorphic to the quotient of the realization of the originalWe give conditions under which the quotient commutes with the nerve functor, as well as conwhich guarantee that the quotient is again a poset. 2004 Elsevier Inc. All rights reserved.

1. Introduction

Assume that we have a finite groupG acting on a posetP in an order-preserving wayThe purpose of this chapter is to compare the various constructions of the quotient,

✩ Research at MSRI was supported in part by NSF grant DMS-9022140. The first author was supporteNSF Postdoctoral Fellowship. The second author was supported by the Swedish Science Council PosFellowship M-PD 11292-303.

* Corresponding author.E-mail addresses:[email protected] (E. Babson), [email protected], [email protected]

(D.N. Kozlov).

0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2001.07.002

440 E. Babson, D.N. Kozlov / Journal of Algebra 285 (2005) 439–450

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ated with this action. Our basic suggestion is to viewP as a category and the group actias a functor fromG to Cat. Then, it is natural to defineP/G to be the colimit of thisfunctor. As a resultP/G is in general a category, not a poset.

After getting a hand on the formal setting in Section 2 we proceed in Section 3imposing different conditions on the group action. We give conditions for each of thlowing properties to be satisfied:

(1) the morphisms ofP/G are exactly the orbits of the morphisms ofP , we call it regu-larity;

(2) the quotient construction commutes with Quillen’s nerve functor;(3) P/G is again a poset.

Furthermore, we study the class of categories which can be seen as the “quotiesure” of the set of all finite posets: loopfree categories.

In Section 4 we draw connections to determining the multiplicity of the trivial charain the induced representations ofG on the homology groups of the nerve of the categderive a formula for the Möbius function ofP/G and, based on formulae of Sundaram aWelker [18], give a quotient analog of Goresky–MacPherson formulae.

As further examples where these methods proved to be essential we would like totion the computation of the homology groups of the deleted symmetric join of an insimplex, see [1], as well as the study of quotients of various partition posets, see [9the earlier work [7,8,16,17].

2. Formalization of group actions and the main question

2.1. Preliminaries

For a small categoryK denote the set of its objects byO(K) and the set of its morphismby M(K). For everya ∈ O(K) there is exactly one identity morphism which we denida , this allows us to identifyO(K) with a subset ofM(K). If m is a morphism ofK froma to b, we writem ∈ MK(a, b), ∂ •m = a and∂ •m = b. The morphismm has an inversem−1 ∈ MK(b, a), if m ◦ m−1 = ida andm−1 ◦ m = idb. If only the identity morphismshave inverses inK thenK is said to be a category without inverses.

We denote the category of all small categories byCat. If K1,K2 ∈ O(Cat) we de-note byF(K1,K2) the set of functors fromK1 to K2. We need three full subcategoriof Cat: P the category of posets (which are categories with at most one morphism, de(x → y), between any two objectsx, y), L the category of loopfree categories (see Dinition 3.9), andGrp the category of groups (which are categories with a single elemmorphisms given by the group elements and the law of composition given by grouptiplication). Finally,1 is the terminal object ofCat, that is, the category with one elemeand one (identity) morphism. The other two categories we use areTop, the category otopological spaces, andSS, the category of simplicial sets.

We are also interested in the functors∆ : Cat → SS andR : SS → Top. The composi-tion is denoted∆̃ : Cat → Top. Here,∆ is the nerve functor, see [13–15]. In particulthe simplices of∆(K) are chains of morphisms inK , with degenerate simplices co

E. Babson, D.N. Kozlov / Journal of Algebra 285 (2005) 439–450 441

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responding to chains that include identity morphisms, see [5,19].R is the topologicalrealization functor, see [11].

Note that both∆ and∆̃ have weak homotopy inverses, i.e., functorsξ : SS → Cat andξ̃ : Top → Cat such that∆̃ ◦ ξ̃ is homotopic to the identity, andR ◦ ∆ ◦ ξ is homotopicto R, see [4].

We recall here the definition of a colimit (see [10,12]).

Definition 2.1. Let K1 andK2 be categories andX ∈ F(K1,K2). A sink of X is a pairconsisting ofL ∈ O(K2), and a collection of morphisms{λs ∈ MK2(X(s),L)}s∈O(K1),such that ifα ∈ MK1(s1, s2) thenλs2 ◦ X(α) = λs1. (One way to think of this collectionof morphisms is as a natural transformation between the functorsX andX′ = X1 ◦ X2,whereX2 is the terminal functorX2 : K1 → 1 andX1 : 1 → K2 takes the object of1 to L.)When(L, {λs}) is universal with respect to this property we call it thecolimit of X andwrite L = colimX.

2.2. Definition of the quotient and formulation of the main problem

Our main object of study is described in the following definition.

Definition 2.2. We say that a groupG acts ona categoryK if there is a functorAK :G →Cat which takes the unique object ofG to K . The colimit ofAK is called thequotientofK by the action ofG and is denoted byK/G.

To simplify notations, we identifyAKg with g itself. Furthermore, in Definition 2.the categoryCat can be replaced with any categoryC, thenK,K/G ∈ O(C). Importantspecial case isC = SS. It arises whenK ∈ O(Cat) and we consider colim∆ ◦ AK =∆(K)/G.

Main Problem. Understand the relation between the topological and the categoricatients, that is, between∆(K/G) and∆(K)/G.

To start with, by the universal property of colimits there exists a canonical sution λ :∆(K)/G → ∆(K/G). In the next section we give combinatorial conditions unwhich this map is an isomorphism.

The general theory tells us that ifG acts on the categoryK , then the colimitK/G exists,sinceCat is cocomplete. We shall now give an explicit description.

An explicit description of the categoryK/G

Whenx is a morphism ofK , denote byGx the orbit ofx under the action ofG. WehaveO(K/G) = {Ga | a ∈ O(K)}. The situation with morphisms is more complicateDefine a relation↔ on the setM(K) by settingx ↔ y, iff there are decompositionx = x1 ◦ · · · ◦ xt and y = y1 ◦ · · · ◦ yt with Gyi = Gxi for all i ∈ [t]. The relation↔is reflexive and symmetric sinceG has identity and inverses, however it is not in genetransitive. Let∼ be the transitive closure of↔, it is clearly an equivalence relation. Denothe ∼ equivalence class ofx by [x]. Note that∼ is the minimal equivalence relation o


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l map

), thisl, butee ofions

meherited

Fig. 1.

M(K) closed under theG action and under composition; that is, witha ∼ ga for anyg ∈ G, and if x ∼ x′ andy ∼ y′ andx ◦ x′ andy ◦ y′ are defined thenx ◦ x′ ∼ y ◦ y′. Itis not difficult to check that the set{[x] | x ∈ M(K)} with the relations∂ •[x] = [∂ •x],∂ •[x] = [∂ •x] and[x] ◦ [y] = [x ◦ y] (whenever the compositionx ◦ y is defined), are themorphisms of the categoryK/G.

Note that ifP is a poset with aG action, the quotient taken inCat need not be a poseand hence may differ from the poset quotient.

Example 2.3. Let P be the center poset in Fig 1. LetS2 act onP by simultaneouslypermutinga with b and c with d . (I) showsP/S2 in P and (II) showsP/S2 in Cat.Note that in this case the quotient inCat commutes with the functor∆ (the canonicasurjectionλ is an isomorphism), whereas the quotient inP does not.

3. Conditions on group actions

3.1. Outline of the results and surjectiveness of the canonical map

In this section we consider combinatorial conditions for a groupG acting on a categorK which ensure that the quotient by the group action commutes with the nerve fuIf AK :G → Cat is a group action on a categoryK then∆ ◦ AK :G → SS is the associ-ated group action on the nerve ofK . It is clear that∆(K/G) is a sink for∆ ◦ AK , andhence, as previously mentioned, the universal property of colimits gives a canonicaλ :∆(K)/G → ∆(K/G). We wish to find conditions under whichλ is an isomorphism.

First we prove in Proposition 3.1 thatλ is always surjective. Furthermore,Ga = [a]for a ∈ O(K), which means that, restricted to 0-skeleta,λ is an isomorphism. If the twosimplicial spaces were simplicial complexes (only one face for any fixed vertex setwould suffice to show isomorphism. Neither one is a simplicial complex in generawhile the quotient of a complex∆(K)/G can have simplices with fairly arbitrary facsets in common,∆(K/G) has only one face for any fixed edge set, since it is a nerva category. Thus forλ to be an isomorphism it is necessary and sufficient to find conditunder which

(1) λ is an isomorphism restricted to 1-skeleta;(2) ∆(K)/G has only one face with any given set of edges.

We will give conditions equivalent toλ being an isomorphism, and then give sostronger conditions that are often easier to check, the strongest of which is also inby the action of any subgroupH of G acting onK .


e

-

Fig. 2.

First note that a simplex of∆(K/G) is a sequence([m1], . . . , [mt ]), mi ∈ M(K), with∂ •[mi−1] = ∂ •[mi], which we will call achain. On the other hand a simplex of∆(K)/G

is an orbit of a sequence(n1, . . . , nt ), ni ∈ M(K), with ∂ •ni−1 = ∂ •ni , which we denoteG(n1, . . . , nt ). The canonical mapλ is given byλ(G(n1, . . . , nt )) = ([n1], . . . , [nt ]).

Proposition 3.1. Let K be a category andG a group acting onK . The canonical mapλ : ∆(K)/G → ∆(K/G) is surjective.

Proof. By the above description ofλ it suffices to fix a chain([m1], . . . , [mt ]) and finda chain(n1, . . . , nt ) with [ni] = [mi]. The proof is by induction ont . The caset = 1 isobvious, just taken1 = m1.

Assume now that we have foundn1, . . . , nt−1, so that[ni] = [mi], for i = 1, . . . , t − 1,andn1, . . . , nt−1 compose, i.e.,∂ •ni = ∂ •ni+1, for i = 1, . . . , t − 2. Since[∂ •nt−1] =[∂ •mt−1] = [∂ •mt ], we can findg ∈ G, such thatg∂ •mt = ∂ •nt−1. If we now takent =gmt , we see thatnt−1 andnt compose, and[nt ] = [mt ], which provides a proof for thinduction step. �3.2. Conditions for injectiveness of the canonical projection

Definition 3.2 (see Fig.2). Let K be a category andG a group acting onK . We say thatthis action satisfiesCondition (R) if the following is true: Ifx, ya, yb ∈ M(K), ∂ •x =∂ •ya = ∂ •yb andGya = Gyb, thenG(x ◦ ya) = G(x ◦ yb).

We say in such case thatG actsregularlyonK .

Proposition 3.3. LetK be a category andG a group acting onK . This action satisfies Condition (R) iff the canonical surjectionλ :∆(K)/G → ∆(K/G) is injective on1-skeleta.

Proof. The injectiveness ofλ on 1-skeleta is equivalent to requiring thatGm = [m], forall m ∈M(K), while Condition (R) is equivalent to requiring thatG(m◦Gn) = G(m◦n),for all m,n ∈ M(K) with ∂ •m = ∂ •n; herem ◦ Gn means the set of allm ◦ gn for whichthe composition is defined.

Assume thatλ is injective on 1-skeleta. The we have the following computation:

G(m ◦ Gn) = Gm ◦ Gn = [m] ◦ [n] = [m ◦ n] = G(m ◦ n),

hence Condition (R) is satisfied.


its

ato-

hted

Fig. 3.

Reversely, assume that Condition (R) is satisfied, that isG(m ◦ Gn) = G(m ◦ n). Sincethe equivalence class[m] is generated byG and composition, it suffices to show that orbare preserved by composition, which is preciselyG(m ◦ Gn) = G(m ◦ n). �

The following theorem is the main result of this chapter. It provides us with combinrial conditions which are equivalent toλ being an isomorphism.

Theorem 3.4 (see Fig. 3). LetK be a category andG a group acting onK . The followingtwo assertions are equivalent for anyt � 2:

(1t ) Condition(Ct ): If m1, . . . ,mt−1,ma,mb ∈ M(K) with ∂ •mi = ∂ •mi−1 for all 2 �i � t −1, ∂ •ma = ∂ •mb = ∂ •mt−1, andGma = Gmb, then there is someg ∈ G suchthatgma = mb andgmi = mi for 1� i � t − 1.

(2t ) The canonical surjectionλ :∆(K)/G → ∆(K/G) is injective ont-skeleta.

In particular, λ is an isomorphism iff(Ct ) is satisfied for allt � 2. If this is the case, wesay that Condition(C) is satisfied.

Proof. (1t ) is equivalent toG(m1, . . . ,mt ) = G(m1, . . . ,mt−1,Gmt); this notation isused, as before, for all sequences(m1, . . . ,mt−1, gmt ) which are chains, that is for whicm1 ◦ · · · ◦ mt−1 ◦ gmt is defined.(2t ) implies Condition (R) above, and so can be restaasG(m1, . . . ,mt ) = (Gm1, . . . ,Gmt).

(2t ) ⇒ (1t ). G(m1, . . . ,mt ) = (Gm1, . . . ,Gmt) = G(Gm1, . . . ,Gmt)

⊇ G(m1, . . . ,mt−1,Gmt) ⊇ G(m1, . . . ,mt ).

(12) ⇒ (22). G(m1,m2) = G(m1,Gm2)

= {g1(m1, g2m2) | ∂ •m1 = ∂ •g2m2

}

= {(g1m1, g2m2) | ∂ •g1m1 = ∂ •g2m2

} = (Gm1,Gm2).

(1t ) ⇒ (2t ), t � 3. We use induction ont .


-in.

rms in

at

t

n.

r

G(m1, . . . ,mt ) = G(m1, . . . ,mt−1,Gmt)

= {(gm1, . . . , gmt−1, g̃mt | ∂ •gmt−1 = ∂ •g̃mt

)}

= {(g1m1, . . . , gtmt ) | ∂ •gimi = ∂ •gi+1mi+1, i ∈ [t − 1]}

= (Gm1, . . . ,Gmt). �Example 3.5. A group action which satisfies Condition(Ct ), but does not satisfy Condition (Ct+1). Let Pt+1 be the order sum oft + 1 copies of the 2-element antichaThe automorphism group ofPt+1 is the direct product oft + 1 copies ofZ2. TakeG to bethe index 2 subgroup consisting of elements with an even number of nonidentity tethe product.

The following condition implies Condition (C), and is often easier to check.

Condition (S). There exists a set{Sm}m∈M(K), Sm ⊆ Stab(m), such that

(1) Sm ⊆ S∂ •m ⊆ Sm′ , for anym′ ∈M(K), such that∂ •m′ = ∂ •m;(2) S∂ •m acts transitively on{gm | g ∈ Stab(∂ •m)}, for anym ∈ M(K).

Proposition 3.6. Condition(S) implies Condition(C).

Proof. Letm1, . . . ,mt−1,ma,mb andg be as in Condition (C), then, sinceg ∈ Stab(∂ •ma),there must exist̃g ∈ S∂ •ma such thatg̃(ma) = mb. From (1) above one can conclude thg̃(mi) = mi , for i ∈ [t − 1]. �

We say that thestrongCondition (S) is satisfied if Condition (S) is satisfied withSa =Stab(a). Clearly, in such a case part (2) of the Condition (S) is obsolete.

Example 3.7. A group action satisfying Condition(S), but not the strong Condition(S).Let K = Bn, lattice of all subsets of[n] ordered by inclusion, and letG = Sn act onBn bypermuting the ground set[n]. Clearly, forA ⊆ [n], we have Stab(A) = SA ×S[n]\A, where,for X ⊆ [n], SX denotes the subgroup ofSn which fixes elements of[n] \ X and acts asa permutation group on the setX. SinceA > B meansA ⊃ B, condition (1) of (S) is nosatisfied forSA = Stab(A): SA × S[n]\A �⊇ SB × S[n]\B . However, we can setSA = SA.It is easy to check that for this choice of{SA}A∈Bn

Condition (S) is satisfied.

We close the discussion of the conditions stated above by the following propositio

Proposition 3.8.

(1) The sets of group actions which satisfy Condition(C) or Condition(S)are closed undetaking the restriction of the group action to a subcategory.

(2) Assume a finite groupG acts on a posetP , so that Condition(S) is satisfied. Letx ∈ P

andSx ⊆ H ⊆ Stab(x), then Condition(S) is satisfied for the action ofH onP�x .


f

ce

nt of aet?

to-

hat allgory

mpleposet.

(3) Assume a finite groupG acts on a categoryK , so that Condition(S) is satisfied withSa = Stab(a) (strong version), andH is a subgroup ofG. Then the strong version oCondition(S) is again satisfied for the action ofH onK .

Proof. (1) and (3) are obvious. To show (2) observe that fora � x we haveSa ⊆ Sx ⊆ H ,henceSa ⊆ H ∩ Stab(a). Thus condition (1) remains true. Condition (2) is true sin{g(b) | g ∈ Stab(a)} ⊇ {g(b) | g ∈ Stab(a) ∩ H }. �3.3. Conditions for the categories to be closed under taking quotients

Next, we are concerned with finding out what categories one may get as a quotieposet by a group action. In particular, we ask:in which cases is the quotient again a posTo answer that question, it is convenient to use the following class of categories.

Definition 3.9. A category is calledloopfreeif it has no inverses and no nonidentity aumorphisms.

Intuitively, one may think of loopfree categories as those which can be drawn so tnontrivial morphisms point down. To familiarize us with the notion of a loopfree catewe make the following observations:

• K is loopfree iff for anyx, y ∈ O(K), x �= y, only one of the setsMK(x, y) andMK(y, x) is nonempty andMK(x, x) = {idx};

• a poset is a loopfree category;• a barycentric subdivision of an arbitrary category is a loopfree category;• a barycentric subdivision of a loopfree category is a poset;• if K is a loopfree category, then there exists a partial order� on the setO(K) such

thatMK(x, y) �= ∅ impliesx � y.

Definition 3.10. SupposeK is a small category, andT ∈ F(K,K). We say thatT is hori-zontalif for any x ∈ O(K), if T (x) �= x, thenMK(x,T (x)) = MK(T (x), x) = ∅. Whena groupG acts onK , we say that the action is horizontal if eachg ∈ G is a horizontalfunctor.

WhenK is a finite loopfree category, the action is always horizontal. Another exaof horizontal actions is given by rank preserving action on a (not necessarily finite)We have the following useful property:

Proposition 3.11. Let P be a finite loopfree category andT ∈ F(P,P ) be a hori-zontal functor. LetT̃ ∈ F(∆(P ),∆(P )) be the induced functor, i.e.,̃T = ∆(T ). Then∆(PT ) = ∆(P )

T̃, wherePT denotes the subcategory ofP fixed byT and∆(P )

T̃denotes

the subcomplex of∆(P ) fixed byT̃ .


he

ts underhave:

in apfree

ry is

f

ndi-

Proof. Obviously,∆(PT ) ⊆ ∆(P )T̃

. On the other hand, if for somex ∈ ∆(P ) we haveT̃ (x) = x, then the minimal simplexσ , which containsx, is fixed as a set and, since torder of simplices is preserved byT , σ is fixed byT pointwise, thusx ∈ ∆(PT ). �

The class of loopfree categories can be seen as the closure of the class of posethe operation of taking the quotient by a horizontal group action. More precisely, we

Proposition 3.12. The quotient of a loopfree category by a horizontal action is agaloopfree category. In particular, the quotient of a poset by a horizontal action is a loocategory.

Proof. Let K be a loopfree category and assumeG acts onK horizontally. First observethatMK/G([x]) = {id[x]}. Because ifm ∈ MK/G([x]), then there existx1, x2 ∈ O(K),m̃ ∈ MK(x1, x2), such that[x1] = [x2], [m̃] = m. Thengx1 = x2 for someg ∈ G, hence,sinceg is a horizontal functor,x1 = x2 and sinceK is loopfree we get̃m = idx1.

Let us show that for[x] �= [y] at most one of the setsMK/G([x], [y]) andMK/G([y], [x])is nonempty. Assume the contrary and pickm1 ∈ MK/G([x], [y]), m2 ∈ MK/G([y], [x]).Then there existx1, x2, y1, y2 ∈ O(K), m̃1 ∈ MK(x1, y1), m̃2 ∈ MK(y2, x2) such that[x1] = [x2] = [x], [y1] = [y2] = [y], [m̃1] = [m1], [m̃2] = [m2]. Chooseg ∈ G such thatgy1 = y2. Then [gx1] = [x2] = [x] and we havegm̃1 ∈ MK(gx1, y2), so m̃2 ◦ gm̃1 ∈MK(gx1, x2). SinceK is loopfree we conclude thatgx1 = x2, but then bothMK(x2, y2)

andMK(y2, x2) are nonempty, which contradicts to the fact thatK is loopfree. �Next, we shall state a condition under which the quotient of a loopfree catego

a poset.

Proposition 3.13. Let K be a loopfree category and letG act onK . The following twoassertions are equivalent:

(1) Condition(SR): If x, y ∈M(K), ∂ •x = ∂ •y andG∂ •x = G∂ •y, thenGx = Gy.(2) G acts regularly onK andK/G is a poset.

Proof. (2) ⇒ (1). Follows immediately from the regularity of the action ofG and the factthat there must be only one morphism between[∂ •x](= [∂ •y]) and[∂ •x](= [∂ •y]) (seeFig. 4).

(1) ⇒ (2). Obviously (SR)⇒ (R), hence the action ofG is regular. Furthermore, ix, y ∈ M(K) and there existg1, g2 ∈ G such thatg1∂

•x = ∂ •y and g2∂ •x = ∂ •y,then we can replacex by g1x and reduce the situation to the one described in Cotion (SR), namely that∂ •x = ∂ •y. Applying Condition (SR) and acting withg−1

1 yieldsthe result. �

Fig. 4.


t

,

Fig. 5.

WhenK is a poset, Condition (SR) can be stated in simpler terms.

Condition (SRP). If a, b, c ∈ K , such thata � b, a � c and there existsg ∈ G such thatg(b) = c, then there exists̃g ∈ G such thatg̃(a) = a andg̃(b) = c (see Fig. 5).

That is, for anya, b ∈ P , such thata � b, we require that the stabilizer ofa acts transi-tively onGb.

Proposition 3.14. Let P be a poset and assumeG acts onP . The action ofG on P

induces an action on the barycentric subdivision BdP (the poset of all chains ofP orderedby inclusion). This action satisfies Condition(S), hence it is regular and∆(BdP)/G ∼=∆((BdP)/G). Moreover, if the action ofG onP is horizontal, then(BdP)/G is a poset.

Proof. Let us choose chainsb, c and a = (a1 > · · · > at), such thata � b and a � c.Thenb = (ai1 > · · · > ail ), c = (aj1 > · · · > ajl

). Assume also that there existsg ∈ G suchthat g(ais ) = ajs for s ∈ [l]. If g fixes a then it fixes everyai , i ∈ [t], henceb = c andCondition (S) follows.

If, moreover, the action ofG is horizontal, then againais = ajs , for s ∈ [l], henceb = c

and Condition (SRP) follows. �

4. Applications

Let us first state two simple, but nevertheless fundamental, facts.

Proposition 4.1. Assume thatG is a finite group which acts onK , a category withouinverses, so that Condition(C) is satisfied. Thenβi(∆(K/G)) = 〈γi,1〉, whereγi is theinduced representation ofG onHi(∆(K)).

Proof. 〈γi,1〉 = βi(∆(K)/G) = βi(∆(K/G)), where the first equality follows from [3Theorem 1] and the second from Theorem 3.4.�Proposition 4.2. LetK be a finite loopfree category andG a finite group which acts onK ,then

χ(∆(K)/G

) = 1

|G|∑

g∈G

χ(∆(Kg)

),

whereKg denotes the subcategory ofK which is fixed byg.


posi-

ed the

e

tionsould

f theas a

bius

ormu-

Proof.

χ(∆(K)/G

) = 1

|G|∑

g∈G

χ(∆(K)g

) = 1

|G|∑

g∈G

χ(∆(Kg)

),

where the first equality follows from [3, Theorem 2] and the second from Protion 3.11. �

Proposition 4.2 can be nicely restated in combinatorial language. To do this we nefollowing definition.

Definition 4.3. Let K be a category, such that∆(K) has finitely many simplices. We defin

µ(K)def= χ̃ (∆(K)). We callµ(K) theMöbius functionof K .

Clearly this definition generalizes the Möbius function of a poset. Similar definihave been given: most notably (and apparently independently) in [6] and [2]. We wlike to mention that ifK is a finite loopfree category then one has a generalization orecursive formula for the computation of the Möbius function (which is often takendefinition of the Möbius function of a poset):

µ(0̂, x

) = −∑

y∈O(K),y<x

mx,yµ(0̂, y

),

wheremx,y = |MK(x, y)|, µ(0̂, 0̂) = 1 andµ(0̂, 1̂) = µ(K). Here 0̂ and1̂ are adjointterminal and initial objects.

Proposition 4.4. Let K be a finite loopfree category,G a finite group acting onK , suchthat Condition(C) is satisfied. Then

µ(K/G) = 1

|G|∑

g∈G

µ(Kg).

Proof. Follows from Proposition 4.2 and Theorem 3.4 and the definition of the Möfunction. �

As another application we obtain a quotient analog of the Goresky–MacPherson flae.

Proposition 4.5. LetA be a subspace arrangement inCn and letG be a finite group which

acts onA. Assume that the induced action ofG on LA (the intersection lattice ofA)satisfies Condition(C). Then

βn−1−i (MA/G) = βi(LA/G) =∑

x∈L>0̂A /G

βi−dimx−1(∆

((0̂, x

)/Stab(x)

)).


.

ace

98.ersion

1.

inal),

. Soc.

. 4 (2)

8 (8)

ringer-

don,

73,

8)

68) 105–

2) 132–

ts of the

onfigura-

ridge

Proof. Follows from [18, Corollaries 2.8 and 2.10], Theorem 3.4 and Proposition 3.8�Remark 4.6. A similar (though less pretty) formula can be derived for real subsparrangements, cf. [18, Theorems 2.4 and 2.5].

Acknowledgment

We would like to thank Eva-Maria Feichtner for the careful reading of this paper.

References

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Group actions on posets