Alexander Duals of Multipermutohedron Ideals

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 1, February 2014, pp. 1–15. c Indian Academy of Sciences

Alexander duals of multipermutohedron ideals

AJAY KUMAR and CHANCHAL KUMAR

Indian Institute of Science Education and Research Mohali, Knowledge City, Sector 81, SAS Nagar 140 306, India E-mail: [email protected]; [email protected]

MS received 29 August 2012; revised 23 November 2012

Abstract. An Alexander dual of a multipermutohedron ideal has many combinatorial properties. The standard monomials of an Artinian quotient of such a dual correspond bijectively to some λ-parking functions, and many interesting properties of these Artinian quotients are obtained by Postnikov and Shapiro (Trans. Am. Math. Soc. 356 (2004) 3109–3142). Using the multigraded Hilbert series of an Artinian quotient of an Alexander dual of multipermutohedron ideals, we obtained a simple proof of Steck determinant formula for enumeration of λ-parking functions. A combinatorial formula for all the multigraded Betti numbers of an Alexander dual of multipermutohedron ideals are also obtained.

Keywords. Multipermutohedron; Alexander dual; Hilbert series; parking functions.

2000 Mathematics Subject Classiﬁcation. 05E40, 13D02.

1. Introduction The notion of Alexander duality for a squarefree monomial ideals has been extended to all monomial ideals by Miller [4]. Let N be the set of nonnegative integers and R = k[x ,x ,...,x ] k = 1 2 n be the standard polynomial ring over a ﬁeld .Forb (b ,b ,...,b ) ∈ Nn b n xbi ∈ R 1 2 n ,letx be the monomial i=1 i . Consider a monomial ideal n I in the polynomial ring R. Choose a = (a1,a2,...,an) ∈ N so that all the minimal generators of I divide the monomial xa. The Alexander dual I [a] of I with respect to a (Deﬁnition 2.1) is a monomial ideal in the polynomial ring R. Many properties of the Alexander dual I [a] of I are given in Chapter 5 in the book of Miller and Sturmfels [5]. In this paper, we study Alexander duals of multipermutohedron ideals. As permutohedron and multipermutohedron are polytopes having very rich combinatorial properties, it is not surprising that the associated multipermutohedron ideals as well as its Alexander duals also have many interesting combinatorial properties. Let u = n (u1,u2,...,un) ∈ N with 0 ≤ u1

1 2 Ajay Kumar and Chanchal Kumar

πu I(u) =x in the polynomial ring R = k[x1,...,xn], generated by the monomial vertex labels xπu of P(u) is called a permutohedron ideal. The Alexander dual of I( , ,...,n) = (n,n,...,n) the permutohedron ideal 1 2 with respect to n is given by [n] n−|A|+1 [n] I(1, 2,...,n) = i∈A xi :∅= A ⊂[n]. The quotient R/I(1, 2,...,n) [ ] n− is an Artinian k-algebra and dimk(R/I (1, 2,...,n)n ) = (n + 1) 1.Inotherwords, the number of standard monomials in the Artinian quotient R/I(1, 2,...,n)[n] is precisely (n + 1)n−1, which is the number of labeled trees on (n + 1) vertices. Thus the monomial ideal I(1, 2,...,n)[n] is called a tree ideal. The vertices of the first barycentric subdivision of an (n − 1)-simplex can be naturally labeled with the minimal generators of the tree ideal I(1, 2,...,n)[n] and the free resolution of tree ideal supported by the first barycentric subdivision Bd(n−1) of an (n − 1)-simplex n−1 is minimal. n If we consider u = (u1,u2,...,un) ∈ N with 0 ≤ u1 ≤ u2 ≤···≤un and ui = uj for some i = j, then also the polytope P(u) and the ideal I(u) are well-defined. In this case, we call P(u) a multipermutohedron and I(u) a multipermutohedron ideal.Forany integer c ≥ 1, we consider un + c − 1 = (un + c − 1,un + c − 1,...,un + c − 1) and [u +c−1] [u −c+1] the Alexander dual I(u) n .Ifu1 ≥ 1, then the quotient R = R/I(u) n is an [ + − ] Artinian k-algebra and the Alexander dual I(u) un c 1 is again given by un−u|A|+c [u +c−1] I(u) n = xi :∅= A ⊂[n] , i∈A except the generating set need not be minimal. In this case also, one would like to know, what is the dimension dimk(R ) or equivalently, the number of standard monomials in [ − + ] R = R/I(u) un c 1 ? A solution of this problem is known and it lies in counting generalized parking functions. The dimension dimk(R ) equals the number of λ-parking functions, where λ = (λ1,λ2,...,λn); λi = un − ui + c. Using a free resolution of R and its multigraded Hilbert series, we give a simple proof of the Steck determinant formula for counting λ-parking functions (Theorem 2.8). We have obtained combinato- [ + − ] rial formula for all the multigraded Betti numbers of the Alexander dual I(u) un c 1 of the multipermutohedron ideal (Theorem 3.2). As an application, we characterized minimality of the cellular free resolution supported by a labeled polyhedral cell complex m Bd (n−1) spanned by the barycenters of some of the faces of an n − 1-simplex n−1 (Theorem 3.7). Some of the results in this paper are implicit in the work of Postnikov and Shapiro [7]. They considered certain classes of monomial ideals; namely monotone monomial ideals, order monomial ideals, and obtained a formula for the number of standard monomials in their Artinian quotient algebras. They also computed the (coarse) Hilbert series of these quotient algebras and obtained an extension of a formula for enumerating λ-parking functions due to Pitman and Stanley [6]. We have used multigraded (or fine) Hilbert series of the quotient algebra R and recovered Steck determinant formula for counting λ-parking functions. Postnikov and Shapiro also gave combinatorial formula for multigraded Betti [ + − ] numbers of strictly monotone monomial ideals. The monomial ideal I(u) un c 1 is strictly monotone if u1

2. Alexander duals of monomial ideals Let n be a positive integer and [n]={1, 2,...,n}.Let be a simplicial complex on the vertex set [n].TheAlexander dual ∗ of is a simplicial complex on [n] given by ∗ ={A ⊆[n]:[n]−A/∈ }. For any subset A ⊆[n], xA = i∈A xi is a squarefree monomial in the polynomial ring R = k[x1,x2,...,xn] over a ﬁeld k.TheStanleyÐReisner ideal I of the simplicial complex is deﬁned to be the squarefree monomial ideal

I =xA : A is a minimal nonface of in R.NowtheAlexander dual of the squarefree monomial ideal I is deﬁned to be the ∗ Stanley–Reisner ideal I∗ of the Alexander dual . The notion of Alexander duals of a squarefree monomial ideals has been extended to = (b ,b ,...,b ) ∈ Nn b monomial ideals by Miller [4]. Let b 1 2 n .Thenx denotes the n xbi b xbi : b > monomial i=1 i and m denotes the monomial ideal i i 0 in the standard polynomial ring R = k[x1,x2,...,xn] over a ﬁeld k. Consider a monomial ideal I in the polynomial ring R.ThenI has a unique minimal set of monomial generators and all the (monomial) primary components of I are unique. Let A and B be subsets of Nn such that {xb : b ∈ A} be the set of minimal generators of I and {mb : b ∈ B} be the set of (monomial) primary components of I. Thus, we have

I =xb : b ∈ A= {mb : b ∈ B}.

n n Choose a = (a1,a2,...,an) ∈ N such that a − b ∈ N for all b ∈ A.Inotherwords, all the minimal generator xb of I divide xa. Whenever a, b ∈ Nn with a − b ∈ Nn,weset a b ∈ Nn by deﬁning its ith coordinate

ai + 1 − bi, if bi > 0, (a b)i = 0, if bi = 0.

DEFINITION 2.1 The Alexander dual I [a] of the monomial ideal I with respect to a is deﬁned to be the monomial ideal

I [a] = {mab : b ∈ A}.

Equivalently, I [a] =xab : b ∈ B. Remark 2.2. (1) The Alexander dual is indeed a duality in the sense that (I [a])[a] = I. [ ] Also, the Alexander dual (I) 1 of a Stanley–Reisner ideal I with respect to n 1 = (1, 1,...,1) ∈ N is precisely I∗ . Therefore, the notion of Alexander duality of monomial ideals introduced by Miller turns out to be an appropriate generalization. 4 Ajay Kumar and Chanchal Kumar

(2) Let aI be the exponent on the LCM of all minimal generators of the monomial ideal I. Then we define the (tight) Alexander dual I ∗ = I [aI ]. The only inadequacy of this notion is that (I ∗)∗ need not equal I, unlike (I [a])[a] = I . We now define multipermutohedron and associated multipermutohedron ideals. Let n be = (m ,m ,...,m ) ∈ Nl m ≥ n = a positive integer and m 1 2 l such that each i 1and l m s = s = j m ( ) = (u ,u ,...,u ) ∈ Nn i=1 i .Set 0 0and j i=1 i.Letu m 1 2 n such that m u m u the first 1 coordinates be equal to 1 and next 2 coordinates be equal to m1+1 and so on. In other words, u =···=u

[u +c−1] Therefore, the quotient R/(I(u) n ) is an Artinian k-algebra if and only if u1 ≥ 1. Proof. Suppose all the minimal generators of a monomial ideal I in R divides xa. Then, n b a−b [a] for any b = (b1,...,bn) with a−b ∈ N , the monomial x ∈/ I if and only if x ∈ I (see Proposition 5.23 of [5]). Clearly, xa−b is a minimal generator of I [a] precisely when b is maximal. If usi ≥ 1, then taking b = (usi −1)E(0,si−1 +1)+(un +c−1)E(si−1 +1,n) + − − [ + − ] or its permutation, we see that xun c 1 b is a minimal generator of I(u(m)) un c 1 .

Remark 2.4. It follows from Lemma 2.3 that Alexander dual of a multipermutohedron ideal is a sum of special multipermutohedron ideals. In fact, the tree ideal I(1, 2,...,n)[n] is a sum of multipermutohedron ideals I(0,...,0,n) + I(0,...,0,n− 1,n− 1) +···+ I(0, 2, 2,...,2) + I(1, 1,...,1). Alexander duals of multipermutohedron ideals 5

This gives another motivation for studying multipermutohedron ideals.

If u1 ≥ 1, then the multigraded Hilbert series of the Artinian k-algebra [ + − ] [ − + ] R/(I(u) un c 1 ) is the sum of ﬁnitely many standard monomial in R = R/I(u) un c 1 . In order to describe the multigraded Hilbert series of R , we need the notion of λ- parking functions. For more on parking functions or λ-parking functions, we refer to [6, 7, 9]. DEFINITION 2.5

A sequence (p1,p2,...,pn) of positive integers is said to be a parking function of length n if its nondecreasing rearrangement q1 ≤ q2 ≤ ··· ≤ qn satisfy qi ≤ i, ∀i.Forλ = (λ1,λ2,...,λn) with λ1 ≥ λ2 ≥ ··· ≥ λn, a sequence (p1,p2,...,pn) of positive integers is said to be a λ-parking function of length n if its nondecreasing rearrangement q1 ≤ q2 ≤···≤qn satisfy qi ≤ λn−i+1, ∀i. A λ-parking function for λ = (n, n − 1,n− 2,...,1) is a parking function. p p p p = x 1 x 2 ...x n k Lemma 2.6. A monomial x 1 2 n is a standard monomial in the Artinian - [u +c−1] algebra R = R/(I(u) n ) if and only if p + 1 = (p1 + 1,p2 + 1,...,pn + 1) is a λ λ = (λ ,λ ,...,λ ) = (u −u +c, u −u +c,...,u −u +c) -parking function for 1 2 n n 1 n 2 n n . R H(R , ) = p−1, Thus, the multigraded Hilbert series of is given by x p∈n x where

n is the set of all λ-parking functions of length n.Also,dimk(R ) = H(R , 1) =|n|.

Proof. Suppose p + 1 = (p1 + 1,p2 + 1,...,pn + 1) is not a λ-parking function. Thus there is a nondecreasing rearrangement q = (q1,q2,...,qn) of p + 1 such that qj > λn−j+1 = un − un−j+1 + c for some j. Equivalently, there are at least n − j + 1 indices i ,i ,...,i p ≥ u − u + c ≤ r ≤ n − j + 1 2 n−j+1 such that ir n n−j+1 for 1 1. This condition p un−un−j+ +c holds if and only if x is divisible by ( i∈A xi) 1 with A ={i1,i2,...,in−j+1}. p [u +c−1] Hence, x ∈ I(u) n . This shows that p + 1 = (p1 + 1,p2 + 1,...,pn + 1) is not a λ-parking function ⇔ xp is not a standard monomial in R . Since the multigraded Hilbert series is the sum of all standard monomials, the second and third parts of the lemma follows. We now proceed to give a proof of Steck determinant [8] formula for counting λ-parking functions. Consider the ﬁrst barycentric subdivision Bd(n−1) of an n − 1- simplex n−1.AvertexofBd(n−1) corresponds to a nonempty subset A ⊆[n] and un−u|A|+c hence it is naturally labeled with the monomial ( α∈A xα) in R.Also,an(i −1)- dimensional face of Bd(n−1) corresponds to a tuple (A1,A2,...,Ai) of nonempty subsets of [n] with ∅=A0 A1 A2 ··· Ai and the monomial label on this i − 1-face is

⎛ ⎞u −u +c i n |Aj | ⎝ xα⎠ .

j=1 α∈Aj −Aj−1

Thus, Bd(n−1) is a labeled simplicial complex. To each labeled simplicial complex or a labeled polyhedral cell complex X, one can associate a free complex of R-modules [1,2]. 6 Ajay Kumar and Chanchal Kumar

Let Fi−1(X) (or simply Fi−1)bethesetofi − 1-faces of X. Then the associated free complex F∗(X) is given by

δi F∗(X) :···−→Fi −→ Fi−1 −→ · · · −→ F1 −→ F0 −→ 0, (2.1) ν(σ) where Fi = R[−ν(σ)], monomial x is the label on the face σ and δi σ ∈Fi−1(X) is a differential. If F∗(X) is a resolution of R/I(X),whereI(X) is the ideal in R generated by the vertex labels of X, then the multigraded Hilbert series of R/I(X) is given by

dim(X)+1 i H(R/I(X), x) = (−1) H(Fi, x) i=0 dim(X)+1 ν(σ) i x = (−1) . (1 − x1)(1 − x2) ···(1 − xn) i=0 σ ∈Fi−1

The free complex F∗(Bd(n−1)) associated to the ﬁrst barycentric subdivision Bd(n−1) [ + − ] is in fact a free resolution of the quotient R = R/(I(u) un c 1 ) (see [7]). This resolution is usually nonminimal, but it can be used to calculate the multigraded Hilbert series H(R , x) of the quotient R .Wehave n 1 H(R , x) = (− )i n ( − x ) 1 j=1 1 j i=0 (A1,...,Ai )∈Fi−1 ⎡ ⎛ ⎞u −u +c⎤ i n |Aj | ⎣ ⎝ ⎠ ⎦ × xα . (2.2)

j=1 α∈Aj −Aj−1

PROPOSITION 2.7

⎧ ⎛ ⎞⎫ n i t −t ⎨ (λ ) j j−1 ⎬ n−i ⎝ tj ⎠ |n|=(n!) (−1) . ⎩ (tj − tj−1)! ⎭ i=0 0=t0

Proof. From Lemma 2.6, we have

Q(x) |n|=H(R , 1) = lim H(R , x) = lim n , x →1, x →1, ( − xj ) 1..., 1..., j=1 1 xn→1 xn→1 where the polynomial Q(x),inviewofeq.(2.2), is

⎡ ⎛ ⎞u −u +c⎤ n i n |Aj | i ⎣ ⎝ ⎠ ⎦ Q(x)= (−1) xα .

i=0 (A1,...,Ai )∈Fi−1(Bd(n−1)) j=1 α∈Aj −Aj−1 Alexander duals of multipermutohedron ideals 7

Now applying L’Hospital’s rule, we see that ∂nQ(x) | |= 1 . n (− )n ∂x ∂x ...∂x 1 1 2 n x=1

∂nQ(x) In the partial derivative , the term corresponding to the tuple (A1,...,Ai) sur- ∂x1∂x2...∂xn vives only if |Ai|=n. Putting |Aj |=tj ,λj = un − uj + c, and observing that the n! i − (A ,...,Ai) |Aj |=tj number of 1-faces 1 with is precisely i (t −t )! ,wegetthe j=1 j j−1 desired result. (λ )j−i+1 μ = n−i+1 ≤ i ≤ j + μ = j +

|n|=(n!) det[μij ]n×n.

r−j = r (λn−j ) e ≤ r ≤ n e = {e ,...,e } Proof. Let vr j=0 (r−j)! j+1 for 1 and n+1 0, where 1 n n is the standard basis of R . The column vector vr is the r-th column of the n × n matrix [μij ]. Thus

v1 ∧ v2 ∧···∧vn = (det[μij ])e1 ∧ e2 ∧···∧en.

It is a straightforward veriﬁcation that the exterior product v1 ∧ v2 ∧···∧vn equals ⎧ ⎛ ⎞⎫ n i t −t ⎨ (λ ) j j−1 ⎬ n−i ⎝ tj ⎠ (−1) e1 ∧ e2 ∧···∧en. ⎩ (tj − tj−1)! ⎭ i=0 0=t0

Since exterior product is distributive and ei ∧ei = 0, terms in the product v1 ∧v2 ∧···∧vn are obtained by choosing one term from each vector vr so that their product give rise to a multiple of e1 ∧ e2 ∧ ··· ∧ en.For0 ≤ i ≤ n and a tuple (t1,t2,...,ti ) with 0 = t0

Then f1 ∧ f2 ∧···∧fn is clearly equal to ⎛ ⎞ ⎛ ⎞ i t −t i (λt ) j j−1 ⎝ j ⎠ ⎝ tj −tj− −1⎠ (−1) 1 e1 ∧ e2 ∧···∧en. (2.3) (tj − tj−1)! j=1 j=1 i t −t − n−i (− ) j j−1 1 = (− ) ∧ ∧ ··· ∧ As j=1 1 1 and the product v1 v2 vn is obtained by summing quantity (2.3) over all the possible values of i and (t1,t2,...,ti),using Proposition 2.7, we get

(n!)v1 ∧ v2 ∧···∧vn =|n|e1 ∧ e2 ∧···∧en. This completes the proof. 8 Ajay Kumar and Chanchal Kumar

3. Multigraded Betti numbers [ + − ] Postnikov and Shapiro studied the monomial ideal I(u(m)) un c 1 in [7], without refer- ring it as an Alexander dual. They explicitly constructed a ﬁnite free resolution of the so-called monotone monomial ideals and this free resolution is minimal if the monomial [ + − ] ideal is strictly monotone. In particular, the ideal I(u(m)) un c 1 is strictly monotone if m = (1, 1,...,1), or equivalently u1

˜ b βi,b(I) = dimk Hi−1(K (I); k) and βi−1,b(I) ˜ |Supp(b)|−i−1 = dimk H (Kb(I); k); i ≥ 1, where the support Supp(b) ={i : bi > 0} (see Theorem 5.11 of [5]). We will be primarily using lower Koszul simplicial complexes in computing multigraded Betti numbers of the [u +c−1] Alexander dual of multipermutohedron ideal I(u(m)) n . The minimal generators [u +c−1] un−u|A|+c of I(u(m)) n are of the form ( j∈A xj ) ,whereA ⊆[n], |A|=si + 1for 0 ≤ i

[un+c−1] β0,b(I (u(m)) ) = 1

for b = (un − usi +1 + c)E(0,si + 1) or its permutation, where 0 ≤ i

For computing higher Betti numbers, we need the following topological result: Consider (i ) (i ) (it ) {n } = n 1 ∗ n 2 ∗···∗n ij a disjoint family j of simplexes and let 1 2 t be a join of - t nα nj n ≤ j ≤ t k H˜j ( ; k) = δ ( ),j skeleton of -simplex j for 1 .Thendim iα +1 dim , α=1 δ ( ) = t i + (t − ) where i,j be the Kronecker delta and dim j=1 j 1 (see Lemma 3.3 of [3]). Let p, q ∈ N and p ≤ q.Then[p, q] denotes an integral interval {r ∈ N : p ≤ r ≤ q}. We also write (p, q] for [p + 1,q]. In order to describe multigraded Betti numbers of the [ + − ] Alexander dual I(u(m)) un c 1 , we need the following notion. Alexander duals of multipermutohedron ideals 9

DEFINITION 3.1

Let J ={j1,j2,...,jt }⊆[n] with 0 = j0

n [un+c−1] Theorem 3.2. For b ∈ N and i ≥ 1, let βi−1,b(I (u(m)) ) be an i − 1-th multi- [u +c−1] graded Betti number of I(u(m)) n in the degree b.Ifu1 ≥ 1, then the multigraded [u +c−1] Betti numbers βi−1,b(I (u(m)) n ) are given as follows: J ={j ,j ,...,j }∈I∗ (1) For 1 2 t m, # & t $ % jα − jα− − 1 [un+c−1] 1 βi−1,b(J )(I (u(m)) ) = δi−1,dwt(J ), siα −1 − jα−1 α=1

where J ∩ (siα −1,siα ]={jα}.Ifπ is a permutation of b(J ), then

[un+c−1] [un+c−1] βi−1,πb(J )(I (u(m)) ) = βi−1,b(J )(I (u(m)) ). = π (J ) J ∈ I∗ (i − ) π (J ) (2) If b b for any m 1 and any permutation of b , then

[un+c−1] βi−1,b(I (u(m)) ) = 0.

[u +c−1] [u +c−1] Proof. The lower Koszul complex Kb(J )(I (u(m)) n ) of the dual I(u(m)) n of the multipermutohedron ideal I(u(m)) is claimed to be the join of skeletons of simplices

(si − −1) (si − −j −1) (s −j − ) (s −j − ) 1 1 ∗ 2 1 1 ∗···∗ iα−1 α−1 1 ∗···∗ it −1 t−1 1 , j1−1 j2−j1−1 jα −jα−1−1 jt −jt−1−1 {e : j + ≤ ν ≤ j } where simplex jα−jα−1−1 is spanned by vertices ν α−1 1 α . This claim is proved by a straightforward veriﬁcation. Consider the vector = E( ,s ) + E(j ,s ) + E(j ,s ) +···+E(j ,s ). v 0 i1−1 1 i2−1 2 i3−1 t−1 it −1 Then v is the vector

. . . v = (1,...,1, 0,...,0, .1,...,1, 0,...,0, ...... 1,...,1, 0,...,0, . .0,...,0) 10 Ajay Kumar and Chanchal Kumar consisting of exactly t strands of 1’s followed by 0’s together with n − jt zeros at the α j − j s − j end. The length of the -th strand is α α−1 and precisely ﬁrst iα −1 α−1 entries α (J ) = t (λ − )E(j ,j ) of the -th strand are 1’s followed by 0’s. Now, set b α=1 jα 1 α−1 α . Clearly, b (J )+v cannot be bigger than or equal to an exponent of any minimal generator [u +c−1] [u +c−1] of I(u(m)) n . Thus v ∈ Kb(J )(I (u(m)) n ).Letπj be a permutation of the j-th strand of the vector v and let π be the product of the (disjoint) permutations πj for [u +c−1] 1 ≤ j ≤ t.Thenwealsohaveπv ∈ Kb(J )(I (u(m)) n ).Ifv is another vector obtained from v by replacing at least one of the 0 by 1, then b (J ) + πv becomes bigger [u +c−1] than or equal to an exponent of some minimal generator of Kb(J )(I (u(m)) n ).This proves the claim. [ + − ] t (K (J )(I ( ( )) un c 1 )) = (si − jα− − ) + (t − ) = The dimension dim b u m α=1 α−1 1 1 1 j − t (j − s ) − t α=1 α iα−1 1. The multigraded Betti number

[un+c−1] ˜ jt −i−1 [un+c−1] βi−1,b(J )(I (u(m)) ) = dimk(H (Kb(J )(I (u(m)) ); k)) for i ≥ 1.

[u +c−1] Clearly, i − 1 = dwt(b(J )) ⇔ jt − i − 1 = dim(Kb(J )(I (u(m)) n )). Thus from the above result on homology groups of the join of skeletons of simplexes, the ﬁrst part [ + − ] of (1) follows. Since minimal generators of I(u(m)) un c 1 are invariant under a permu- [u +c−1] [u +c−1] tation we have βi,πb(J )(I (u(m)) n ) = βi,b(J )(I (u(m)) n ), for a permutation π of b(J ). n Let b = (b1,b2,...,bn) ∈ N such that b = πb(J ) for any permutation π. Changing b by a permutation, we may assume that b1 ≥ b2 ≥···≥bn. The nonzero Betti numbers b of a monomial ideal exist in a multidegree b only if the monomial x is a LCM of some = t λ E(j ,j ) set of minimal generators of the monomial ideal. Therefore, b α=1 jα α−1 α J ={j ,j ,...,j }∈I∗ J ∈/ I∗ (i − ) for some 1 2 t m. But by the given condition, m 1 . Thus [u +c−1] βi−1,b(I (u(m)) n ) = 0. This proves (2). Remark 3.3. Theorem 3.2 looks quite similar to Theorem 3.5 in [3]. This is not surprising as there is a general duality for Betti numbers of a monomial ideal I and its Alexander dual I [a]. In fact,

[a] βn−i,b(S/I) = βi,a+1−b(I ) for b = (b1,...,bn) with 1 ≤ bi ≤ ai, ∀i (Theorem 5.48 of [5]). However, this duality does not give all the multigraded Betti numbers of the dual I [a].

COROLLARY 3.4 [u +c−1] Let βi−1(I (u(m)) n ) be the i − 1-th Betti number of the Alexander dual I( ( ))[un+c−1] u ≥ J ={j ,j ,...,j }∈I∗ (i − ) u m . Suppose 1 1 and for 1 2 t m 1 , we set $ %$ % t j − j − j βJ = α α−1 1 α+1 , i−1 siα −1 − jα−1 jα α=1

[un+c−1] where J ∩ (siα −1,siα ]={jα} and jt+1 = n.Thenβi−1(I (u(m)) ) = J J ∈I∗ (i− ) β . m 1 i−1 Alexander duals of multipermutohedron ideals 11

Proof. Let Per(b(J )) be the set of all permutations of b(J ). Then, in view of Theorem 3.2, we have [un+c−1] [un+c−1] βi−1(I (u(m)) ) = βi−1,b(I (u(m)) ) b∈Nn ⎡ ⎤ ⎣ [un+c−1] ⎦ = βi−1,πb(J)(I (u(m)) ) J∈I∗ (i− ) π∈ ( (J)) m 1 Per b = βJ , i−1 J∈I∗ (i− ) m 1

J β = β (I ( ( ))[un+c−1]) J ={j ,j ,...,j }∈ where i−1 π∈Per(b(J )) i−1,πb(J ) u m .For 1 2 t I∗ (i − ) J ∩ (s ,s ]={j } m 1 with iα −1 iα α ,wehave t $ % jα − jα− − 1 [un+c−1] 1 βi−1,πb(J )(I (u(m)) ) = , siα −1 − jα−1 α=1 for all π ∈ Per(b(J )). The number of permutations π of b(J ) is t $ % n! jα+1 |Per(b(J ))|= t = . (jα+ − jα)! jα α=1 1 α=1 Therefore, # $ %&# $ %& t j − j − t j βJ = α α−1 1 α+1 . i−1 siα −1 − jα−1 jα α=1 α=1 I˜ ∗ (i − ) ={J ∈ I∗ (i − ) : J ∩ (s ,s ]=∅} u = Remark 3.5. Let m 1 m 1 0 1 . Then for 1 0, [ + − ] the formula for Betti numbers of the Alexander dual I(u(m)) un c 1 as in Theorem 3.2 I∗ (i − ) I˜ ∗ (i − ) or Corollary 3.4 remain valid by just replacing m 1 with m 1 . We have already obtained formula for the zeroth Betti number of the Alexander dual [ + − ] I(u(m)) un c 1 . For the ﬁrst Betti number, we need to determine all dual m-isolated J ∈ I∗ ( ) J ={s + } ≤ α

We have seen that the ﬁrst barycentric subdivision Bd(n−1) of an n−1-simplex n−1 [ + − ] supports a free resolution of the quotient R/I(u(m)) un c 1 of the Alexander dual of the m multipermutohedron ideal. Now consider a polyhedral cell complex Bd (n−1) obtained by modifying the ﬁrst barycentric subdivision Bd(n−1) as follows: First assume that m u1 ≥ 1. In this case, the vertices of the polyhedral cell complex Bd (n−1) are precisely the barycenters corresponding to the subsets A ⊆[n] with |A|=si + 1for0≤ i

If u1 = 0, then deleting all the edges containing the vertices of n − 1-simplex n−1,the combinatorial formula takes the form $ %$ % n s + m ω 1 f1(Bd (n−1)) = sω + 1 sν + 1 0<ν<ω

A combinatorial formula for the higher dimensional faces of the polyhedral cell complex m m Bd (n−1) are quite cumbersome. But if m = (m1, 1,...,1),thenfi(Bd (n−1)) can be easily calculated.

Theorem 3.6. Let m = (m1, 1,...,1).Then

m [un+c−1] fi−1(Bd (n−1)) = βi−1(I (u(m)) ) ∀i ≥ 1.

Proof. Firstly we consider the case u1 ≥ 1. We know that (i−1)-faces of the ﬁrst barycentric subdivision Bd(n−1) correspond to a chain of nonempty subsets of [n] of length i. Since the barycenters corresponding to the subsets A ⊂[n] with 1 < |A|≤m1 are m missing, an (i − 1)-face of the polyhedral complex Bd (n−1) corresponds to a chain

A1 A2 ··· At

m of subsets of [n] such that either all Ai’s represent vertices of Bd (n−1) or 1 < |A1|≤ m1 < |A2|. In the former case, t = i, while in the latter case, t = i −|A1|+1 as it represents the (i − 1)-face spanned by vertices of n−1, corresponding to sin- gleton subsets of A1, and the barycenters A2,...,At .Let|Ai|=ji.ThenJ = {j1,j2,...,jt } is a dual m-isolated subset with dwt(J) = j1 + (t − 1) − 1 = i − 1, m and every (i − 1)-face of Bd (n−1) arises in this way. In this case, all the faces of m m Bd (n−1) are simplicial and thus Bd (n−1) is a (n − 1)-dimensional simplicial complex. J m Let f be the number of (i − 1)-faces of Bd (n− ) associated to a dual m-isolated i−1 1 ∗ J t jα+ J ∈ I (i − ) i − f = 1 jt+ = subset m 1 with dual weight 1. Then i−1 α=1 jα ,where 1 t j n = (m , ,..., ) βJ = α+1 .Form 1 1 1 , using Corollary 3.4, i−1 α=1 jα , because either jα+1 − jα − 1 = 0orsiα −1 − jα = 0. Thus f ( m( )) = f J i−1 Bd n−1 i−1 J ∈I∗ (i−1) m J [ + − ] = β = β (I ( ( )) un c 1 ). i−1 i−1 u m J ∈I∗ (i− ) m 1 m If u1 = 0, then vertices of the n − 1-simplex n−1 are no longer vertices of Bd (n−1). m Thus an (i − 1)-face of Bd (n−1) corresponds to a chain

A1 A2 ... At of subsets of [n] with t = i and |A1| >m1. Clearly, maximal such chain has length m n − m1 and hence the dimension of Bd (n−1) is n − m1 − 1. In this case, we have f ( m( )) = f J i−1 Bd n−1 i−1 J ∈I˜ ∗ (i−1) m J [ + − ] = β = β (I ( ( )) un c 1 ). i−1 i−1 u m J ∈I˜ ∗ (i− ) m 1 This completes the proof. 14 Ajay Kumar and Chanchal Kumar

m AvertexofBd (n−1) corresponds to a nonempty subset A ⊆[n], and it is nat- un−u|A|+c m urally labeled with the monomial ( i∈A xi) . Thus Bd (n−1) is a labeled polyhedral cell complex. The free complex associated to the labeled polyhedral cell com- m [u +c−1] plex Bd (n−1) gives a cellular free resolution of the ideal I(u(m)) n .Wenow m investigate minimality of the cellular resolution supported by Bd (n−1) . m Theorem 3.7. The cellular resolution supported by Bd (n−1) is the minimal resolution [ + − ] of I(u(m)) un c 1 if and only if mα = 1 for 2 ≤ α ≤ l. Proof. Suppose the free resolution supported by the labeled polyhedral cell m [u +c−1] complex Bd (n−1) minimally resolves I(u(m)).Thenβ1(I (u(m)) n ) = m m f1(Bd (n−1)), the number of edges of Bd (n−1). Using equations (3.1)and(3.2), and similar equations for the case u1 = 0, we see that there are at most one α with sα+1 − sα ≥ 2; namely α = 0ifu1 ≥ 1 and no such α if u1 = 0. Thus in either case, mα+1 = sα+1 − sα = 1forα ≥ 1. This proves the direct part. Conversely, let mα = 1forα ≥ 2. Then the cellular free resolution sup- m ported by the labeled polyhedral cell complex Bd (n−1) is minimal, because [u +c−1] m βi−1(I (u(m)) n ) = fi−1(Bd (n−1)) ∀i ≥ 1, in view of Theorem 3.6. Remark 3.8. In [3], it is proved that the cellular resolution supported by the multipermutohedron P(u(m)) is the minimal resolution of the multipermutohedron ideal I(u(m)) if and only if mα = 1for2 ≤ α ≤ l. In spite of the identical resemblance, in view of Remark 3.3, Theorem 3.7 about the minimal resolution of Alexander dual [ + − ] I(u(m)) un c 1 can not simply be deduced from the minimal resolution of I(u(m)). At the end, we give some examples of cellular free resolutions of the Alexander duals of multipermutohedron ideals for n = 3. (1) Let u(m) = (a,a,b), 0

c c c ¯ (xy) ,(xz) ,(yz) . The dual m-isolated subsets in this case are J0 ={1}, J0 ={2},J1 = J { , }, J¯ ={} J ={, } J J¯ i β 0 = 1 2 1 3 and 2 1 3 ,where i (or i) has dual weight .Wehave 0 J¯ J J¯ J ,β 0 = ,β 1 = ,β 1 = β 2 = I(a,a,b)[b+c−1] 3 0 3 1 6 1 2and 2 3. Thus the Betti numbers of m are β0 = 6,β1 = 8andβ2 = 3. Since the number of edges of Bd (2) is 9 > 8 = β1,the free complex associated with this labeled simplicial complex is a nonminimal resolution of the Alexander dual I(a, b, b)[b+c−1].

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