Euler characteristic reciprocity for chromatic and order polynomials 6 Takahiro Hasebe∗, Masahiko Yoshinaga† 1 0 2 January 5, 2016 n a J 3 Abstract ] O The Euler characteristic of a semialgebraic set can be considered C as a generalization of the cardinality of a finite set. An advantage of . h semialgebraic sets is that we can define “negative sets” to be the sets t a with negative Euler characteristics. Applying this idea to posets, we m introducethenotion ofsemialgebraic posets. Using“negative posets”, [ we establish Stanley’s reciprocity theorems for chromatic and order 1 polynomials at the level of Euler characteristics. v 4 5 Contents 2 0 0 1 Introduction 2 . 1 0 2 Semialgebraic posets and Euler characteristics 5 6 1 2.1 Semialgebraic sets . . . . . . . . . . . . . . . . . . . . . . . . . 5 : v 2.2 Semialgebraic posets . . . . . . . . . . . . . . . . . . . . . . . 5 i X 3 Euler characteristic reciprocity 7 r a 3.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Proof of the main results . . . . . . . . . . . . . . . . . . . . . 10 3.3 Stanley’s reciprocity for order polynomials . . . . . . . . . . . 12 ∗DepartmentofMathematics,HokkaidoUniversity,North10,West8,Kita-ku,Sapporo 060-0810,JAPAN E-mail: [email protected] †DepartmentofMathematics,HokkaidoUniversity,North10,West8,Kita-ku,Sapporo 060-0810,JAPAN E-mail: [email protected] 1 4 Chromatic polynomials for finite graphs 15 4.1 Chromatic polynomials and Euler characteristics . . . . . . . . 15 4.2 Acyclic orientations . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Euler characteristic reciprocity for chromatic polynomials . . . 17 1 Introduction Let P be a finite poset. The order polynomial O≤(P,t) ∈ Q[t] and the strict order polynomial O<(P,t) ∈ Q[t] are polynomials which satisfy O≤(P,n) = #Hom≤(P,[n]), (1) O<(P,n) = #Hom<(P,[n]), where [n] = {1,...,n} with normal ordering and Hom≤(<)(P,[n]) = {f : P −→ [n] | x < y =⇒ f(x) ≤ (<)f(y)} is the set of increasing (resp. strictly increasing) maps. These two polynomials are related to each other by the following reci- procity theorem proved by Stanley ([9, 10], see also [1, 3, 4] for a recent survey). O<(P,t) = (−1)#P ·O≤(P,−t). (2) By putting t = n, the formula (2) can be informally presented as follows. “ #Hom<(P,[n]) = (−1)#P ·#Hom≤(P,[−n]). ” (3) It is a natural problem to extend the above reciprocity to homomorphisms between arbitrary (finite) posets P and Q. We may expect a formula of the following type. “ #Hom<(P,Q) = (−1)#P ·#Hom≤(P,−Q). ” (4) Of course this is not a mathematically justified formula. In fact, we do not have the notion of a “negative poset −Q.” The negative poset −Q in the right-hand side does not make sense. In [8], Schanuel discussed what “negative sets” should be. A possible answer is that a negative set is nothing but a semialgebraic set which has a negative Euler characteristic (Table 1). For example, the open simplex σ◦ = {(x ,...,x ) ∈ Rd | 0 < x < ··· < x < 1} d 1 d 1 d 2 Finite set Semialgebraic set Cardinality Euler characteristic Table 1: Negative sets ◦ has the Euler characteristic e(σ ) = (−1)d, and the closed simplex d σ = {(x ,...,x ) ∈ Rd | 0 ≤ x ≤ ··· ≤ x ≤ 1} d 1 d 1 d has e(σ ) = 1. Thus we have the following “reciprocity” d e(σ◦ ) = (−1)d ·e(σ ). (5) d d This formula looks alike Stanley’s reciprocity (2). This analogy would indi- cate that (2) could be explained via thecomputations of Euler characteristics of certain semialgebraic sets. In this paper, by introducing the notion of semialgebraic posets, we set- tle Euler characteristic reciprocity theorems for poset homomorphisms and chromatic functors, which imply Stanley’s reciprocities as corollaries. Semi- algebraic posets also provide a rigorous formulation for (4). Briefly, asemialgebraic posetP isasemialgebraicset withposetstructure such thattheordering is defined semialgebraically (seeDefinition2.2). Finite posets and the open interval (0,1) ⊂ R are examples of semialgebraic posets. A semialgebraic poset P has the Euler characteristic e(P) ∈ Z which is an invariant of semialgebraic structure of P (see §2.1). In particular, if P is a finite poset, then e(P) = #P, and if P is the open interval (0,1), then e((0,1)) = −1. The philosophy presented in the literature [8] suggests to consider the “moduli space” Hom≤(<)(P,Q) of poset homomorphisms from a finite poset P to a semialgebraic poset Q, and then computing the Euler characteristic of the moduli space instead of counting the number of maps. We can naturally expect that the product semialgebraic poset Q×(0,1) could play the role of the negative poset “−Q” since e(Q×(0,1)) = −e(Q). In fact, we have the following result. Theorem 1.1 (Proposition 2.6 and Theorems 3.1, 3.5). Let P be a finite poset, and Q be a semialgebraic poset. (i) Hom≤(P,Q) and Hom<(P,Q) possess structures of semialgebraic sets. (ii) The following reciprocity of Euler characteristics holds, e(Hom<(P,Q)) = (−1)#P ·e(Hom≤(P,Q×(0,1))), (6) e(Hom≤(P,Q)) = (−1)#P ·e(Hom<(P,Q×(0,1))). (7) 3 (iii) Let T be a semialgebraic totally ordered set. Then e(Hom≤(P,T)) = O≤(P,e(T)), (8) e(Hom<(P,T)) = O<(P,e(T)). (9) The formula (6) may be considered as a rigorous formulation of (4). Fur- thermore, by putting Q = [n] and T = [n]×(0,1), we can recover Stanley’s reciprocity (2) from (6) and (8) (see §3.3). Semialgebraic posets are also applicable to chromatic theory of finite graphs. Let G = (V,E) beafinite graph. Foranyset X, we canassociate the graph configuration space χ(G,X) (Definition 4.1). If X is a semialgebraic set, then χ(G,X) is also a semialgebraic set. The chromatic polynomial of G is a polynomial χ(G,t) ∈ Z[t] determined by χ(G,n) = #χ(G,[n]). The next result generalizes a result in [7, Theorem 2], where X was chosen to be a complex projective space. Theorem 1.2 (Theorem 4.2). Let G = (V,E) be a finite graph and X be a semialgebraic set. Then e(χ(G,X)) = χ(G,e(X)). A “negative set” also appears in chromatic polynomials. Stanley estab- lished a reciprocity, showing that (−1)#V ·χ(G,−n) can beinterpreted as the number of pairs of an acyclic orientation and a map V −→ [n] compatible with the orientation ([11]). We can give a meaning of “#χ(G,[−n])” using the above result, e(χ(G,X)) = χ(G,e(X)), when X = [n]×(0,1). A precise setting and results are stated below. LetT beatotallyorderedset. WeintroducethemodulispaceAOC≤(G,T) (resp. AOC<(G,T)) of the pairs consisting of an acyclic orientation on the edges E of G with a compatible (resp. strictly compatible) map V → T (Definition 4.4). When T is a semialgebraic totally ordered set, the moduli spaces AOC≤(G,T) and AOC<(G,T) also admit structures of semialgebraic sets. Part of theEuler characteristic reciprocity can beformulated asfollows. Theorem 1.3 (Theorem 4.5, Corollary4.7). Let G = (V,E) be a finite graph and T be a semialgebraic totally ordered set. Then e(AOC≤(G,T)) = (−1)#V ·e(AOC<(G,T ×(0,1))), (10) e(AOC<(G,T)) = χ(G,e(T)). (11) Byputting T = [n] in(10)andT = [n]×(0,1)in(11), weobtainStanley’s reciprocity (Corollary 4.8) #AOC≤(G,[n]) = (−1)#V ·χ(G,−n). 4 2 Semialgebraic posets and Euler character- istics 2.1 Semialgebraic sets A subset X ⊂ Rn is said to be a semialgebraic set if it is expressed as a Boolean connection (i.e. a set expressed by a finite combination of ∪,∩ and complements) of subsets of the form {x ∈ Rn | p(x) > 0}, where p(x) ∈ R[x ,...,x ] is a polynomial. Let f : X −→ Y be a map 1 n between semialgebraic sets X ⊂ Rn and Y ⊂ Rm. It is called semialgebraic if the graph Γ(f) = {(x,f(x)) | x ∈ X} ⊂ Rm+n is a semialgebraic set. If f is semialgebraic then the pull-back f−1(Y) and the image f(X) are also semialgebraic sets (see [2, 5] for details). Any semialgebraic set X has a finite partition into Nash cells, namely, k a partition X = X such that X is Nash diffeomorphic (that is a α=1 α α semialgebraic analyticdiffeomorphism) totheopencell (0,1)dα forsomed ≥ α F 0. Then the Euler characteristic k e(X) := (−1)dα (12) α=1 X is independent from the partition ([6]). Moreover, the Euler characteristic satisfies e(X ⊔Y) = e(X)+e(Y), e(X ×Y) = e(X)×e(Y). Example 2.1. As mentioned in §1, the closed simplex σ and the open d ◦ ◦ simplex σ have e(σ ) = 1 and e(σ ) = (−1)d. d d d 2.2 Semialgebraic posets Definition 2.2. (P,≤) is called a semialgebraic poset if (1) (P,≤) is a partially ordered set, and 5 (2) there is an injection i : P ֒→ Rn (n ≥ 0) such that the image i(P) is a semialgebraic set and the image of {(x,y) ∈ P ×P | x ≤ y}, by the map i×i : P ×P −→ Rn ×Rn, is also a semialgebraic subset of Rn ×Rn. Let P and Q be semialgebraic posets. The set of homomorphisms (strict homomorphisms) of semialgebraic posets is defined by f is a semialgebraic map s.t. Hom≤(<)(P,Q) = f : P −→ Q . (13) x < y =⇒ f(x) ≤ (<)f(y) (cid:26) (cid:12) (cid:27) (cid:12) Example 2.3. (1) A finite poset(cid:12) (P,≤) admits the structure of a semial- (cid:12) gebraic poset, since any finite subset in Rn is a semialgebraic set. A finite poset has the Euler characteristic e(P) = #P. (2) The open interval (0,1) and the closed interval [0,1] are semialgebraic posets with respect to the usual ordering induced from R. Their Euler characteristics are e((0,1)) = −1 and e([0,1]) = 1, respectively. Let P and Q be posets. Recall that the product P × Q admits poset structure by the lexicographic ordering: p < p , or, (p ,q ) ≤ (p ,q ) ⇐⇒ 1 2 1 1 2 2 p = p and q ≤ q , 1 2 1 2 (cid:26) for (p ,q ) ∈ P ×Q. i i Proposition 2.4. Let P and Q be semialgebraic posets. Then the product poset P ×Q (with lexicographic ordering) admits the structure of a semial- gebraic poset. Proof. Suppose P ⊂ Rn and Q ⊂ Rm. Then {((p ,q ),(p ,q )) ∈ (P ×Q)2 | (p ,q ) ≤ (p ,q )} 1 1 2 2 1 1 2 2 = {(p ,q ,p ,q ) ∈ (P ×Q)2 | (p < p ) or (p = p and q ≤ q )} 1 1 2 2 1 2 1 2 1 2 ≃ {(p ,p ) ∈ P2 | p < p }×Q2 ⊔ P ×{(q ,q ) ∈ Q2 | q ≤ q } 1 2 1 2 1 2 1 2 is also (cid:0)semialgebraic since semialgebr(cid:1)aici(cid:0)ty is preserved by disjoint u(cid:1)nion, complement and Cartesian products. Proposition 2.5. Let P and Q be semialgebraic posets. Then the first projection π : P ×Q −→ P is a homomorphism of semialgebraic posets. 6 Proof. This is straightforward from the definition of the lexicographic order- ing. The next result shows that the “moduli space” of homomorphisms from a finite poset to a semialgebraic poset has the structure of a semialgebraic set. Proposition 2.6. Let P be a finite poset and Q be a semialgebraic poset. Then Hom≤(P,Q) and Hom<(P,Q) have structures of semialgebraic sets. Proof. Let us set P = {p ,...,p } and L = {(i,j) | p < p }. Since each 1 n i j element f ∈ Hom≤(P,Q)canbeidentifiedwiththetuple(f(p ),...,f(p )) ∈ 1 n Qn, we have the expression Hom≤(P,Q) ≃ {(q ,...,q ) ∈ Qn | q ≤ q for (i,j) ∈ L} 1 n i j = {(q ,...,q ) ∈ Qn | q ≤ q }. 1 n i j (i,j)∈L \ Clearly, the right-hand side is a semialgebraic set. The semialgebraicity of Hom<(P,Q) is similarly proved. 3 Euler characteristic reciprocity 3.1 Main results We can formulate a reciprocity theorem for semialgebraic posets. Theorem 3.1. Let P be a finite poset and Q be a semialgebraic poset. Then e(Hom<(P,Q)) = (−1)#P ·e(Hom≤(P,Q×(0,1))), (14) and e(Hom<(P,Q×(0,1))) = (−1)#P ·e(Hom≤(P,Q)). (15) Note that Q×(0,1) in the right-hand side of (14) is a semialgebraic poset with Euler characteristic e(Q×(0,1)) = −e(Q). (16) In view of the relation (16), Theorem 3.1 may be considered as a generaliza- tion of Stanley’s reciprocity (see Corollary 3.10). Before the proof of Theorem 3.1, we present an example which illustrates the main idea of the proof. 7 Example 3.2. Let P = Q = {1,2} with 1 < 2. Clearly we have Hom<(P,Q) = {id}. Let us describe Hom≤(P,Q× (0,1)). Note that Q× (0,1) is isomorphic to the semialgebraic totally ordered set (1, 3)⊔(2, 5) by the isomorphism 2 2 3 5 t ϕ : Q×(0,1) −→ 1, ⊔ 2, ,(a,t) 7−→ a+ . 2 2 2 (cid:18) (cid:19) (cid:18) (cid:19) A homomorphism f ∈ Hom≤(P,Q × (0,1)) is described by the two values f(1) = (a ,t ) and f(2) = (a ,t ) ∈ Q × (0,1). The condition imposed on 1 1 2 2 a ,a ,t and t (by the inequality f(1) ≤ f(2)) is 1 2 1 2 (a < a ), or (a = a and t ≤ t ), 1 2 1 2 1 2 which is equivalent to a + t1 ≤ a + t2. Therefore, the semialgebraic set 1 2 2 2 Hom≤(P,Q×(0,1)) can be described as in Figure 1. Each diagonal triangle a < a 1 2 a = a = 2,t ≤ t 1 2 1 2 a = a = 1,t ≤ t 1 2 1 2 Figure 1: f(1) ≤ f(2). ◦ ◦ in Figure 1 has a stratification σ ⊔σ . Therefore the Euler characteristic is 2 1 ◦ ◦ ◦ ◦ e(σ ⊔σ ) = e(σ )+e(σ ) = (−1)2+(−1)1 = 0. Ontheotherhand, thesquare 2 1 2 1 region corresponding to a < a has the Euler characteristic (−1)2 = 1. 1 2 Hence we have e(Hom≤(P,Q×(0,1))) = 1 = e(Hom<(P,Q)). The following lemma will be used in the proof of Theorem 3.1. 8 Lemma 3.3. Let P ⊂ Rn be a d-dimensional polytope (i.e., a convex hull of a finite set). Fix a hyperplane description P = {α ≥ 0}∩···∩{α ≥ 0} 1 N of P where α are affine maps from Rn to R. For a given x ∈ P, define the i 0 associated locally closed subset P of P (see Figure 2) by x0 P = {α ≥ 0}∩ {α > 0}. x0 i i αi(\x0)=0 αi(\x0)>0 Then the Euler characteristic is ◦ (−1)d, if x ∈ P e(P ) = 0 x0 ( 0, otherwise (x0 ∈ ∂P), ◦ ◦ where P is the relative interior of P and ∂P = P rP. x 0 x 0 Figure 2: P . x0 ◦ ◦ ◦ Proof. If x ∈ P, then P = P. The Euler characteristic is e(P) = (−1)d. 0 x0 Suppose x ∈ ∂P. Then P can be expressed as 0 x0 ◦ P = F, (17) x0 FG∋x0 ◦ where F runs over the faces of P containing x and F denotes its relative 0 interior. Then we obtain the decomposition ◦ ◦ P = P ⊔ F. x0 F∋x0G,F⊂∂P ◦ We look at the structure of the second component Z := F. For F∋x0,F⊂∂P anypointy ∈ Z,thesegment[x ,y]iscontainedinZ. HenceZ iscontractible 0 F 9 open subset of ∂P, which is homeomorphic to the (d−1)-dimensional open disk. The Euler characteristic is computed as ◦ e(P ) = e(P)+e(Z) x0 = (−1)d +(−1)d−1 = 0. 3.2 Proof of the main results Now we prove (14) of Theorem 3.1. Let ϕ ∈ Hom<(P,Q×(0,1)). Then ϕ is a pair of maps ϕ = (f,g), where f : P −→ Q and g : P −→ (0,1). Let π : Q×(0,1) −→ Q be the first 1 projection. Since π is order-preserving (Proposition 2.5), so is f = π ◦ϕ, 1 1 and hence f ∈ Hom≤(P,Q). In order to compute the Euler characteristics, we consider the map π : Hom≤(P,Q×(0,1)) −→ Hom≤(P,Q), ϕ 7−→ π ◦ϕ = f. (18) 1∗ 1 Let us set M :=Hom≤(P,Q)rHom<(P,Q) (19) ={f ∈ Hom≤(P,Q) | ∃x < y ∈ P s.t. f(x) = f(y)}. Then obviously, we have Hom≤(P,Q) = Hom<(P,Q)⊔M. (20) This decomposition induces that of Hom≤(P,Q×(0,1)), Hom≤(P,Q×(0,1)) = π−1(Hom<(P,Q))⊔π−1(M). (21) 1∗ 1∗ By the additivity of the Euler characteristics, we obtain e Hom≤(P,Q×(0,1)) = e π−1(Hom<(P,Q)) +e(π−1(M)). (22) 1∗ 1∗ We cla(cid:0)im the following two(cid:1) equa(cid:0)lities which are su(cid:1)fficient for the proof of (14). e π−1(Hom<(P,Q)) = (−1)#P ·e(Hom<(P,Q)) (23) 1∗ e(π−1(M)) = 0 (24) (cid:0) 1∗ (cid:1) 10