Simple Polyominoes are Prime
aa r X i v : . [ m a t h . A C ] J a n SIMPLE POLYOMINOES ARE PRIME
AYESHA ASLOOB QURESHI, TAKAFUMI SHIBUTA AND AKIHIRO SHIKAMA
Abstract.
In this paper we show that polyomino ideal of a simple polyominocoincides with the toric ideal of a weakly chordal bipartite graph and hence it hasa quadratic Gr¨obner basis with respect to a suitable monomial order.
Introduction
Polyominoes are two dimensional objects which are originally rooted in recre-ational mathematics and combinatorics. They have been widely discussed in con-nection with tiling problems of the plane. Typically, a polyomino is plane figureobtained by joining squares of equal sizes, which are known as cells. In connectionwith commutative algebra, polyominoes were first discussed in [8] by assigning eachpolyomino the ideal of its inner 2-minors or the polyomino ideal . The study of idealof t -minors of an m × n matrix is a classical subject in commutative algebra. Theclass of polyomino ideal widely generalizes the class of ideals of 2-minors of m × n matrix as well as the ideal of inner 2-minors attached to a two-sided ladder.Let P be a polyomino and K be a field. We denote by I P , the polyomino idealattached to P , in a suitable polynomial ring over K . The residue class ring definedby I P is denoted by K [ P ]. It is natural to investigate the algebraic properties of I P depending on shape of P . In [8], it was shown that for a convex polyomino, theresidue ring K [ P ] is a normal Cohen-Macaualay domain. More generally, it was alsoshown that polyomino ideals attached to a row or column convex polyomino is also aprime ideal. Later in [2], a classification of the convex polyominoes whose polyominoideals are linearly related is given. For some special classes of polyominoes, theregularirty of polyomino ideal is discussed in [3].In [8], it was conjectured that polyomino ideal attached to a simple polyomino isprime ideal. Roughly speaking, a simple polyomino is a polyomino without ’holes’.This conejcture was further studied in [4], where authors introduced balanced poly-ominoes and proved that polyomino ideals attached to balanced polyominoes areprime. They expected that all simple polyominoes are balanced, which would thenprove simple polyominoes are prime. This question was further discussed in [5],where authors proved that balanced and simple polyominoes are equivalent. Inde-pendent of the proofs given in [5], in this paper we show that simple polyominoesare prime by identifying the attached residue class ring K [ P ] with the edge rings ofweakly chordal graphs. Moreover, from [6], it is known that toric ideal of the edge Mathematics Subject Classification.
Key words and phrases. polyominoes, toric ideals. ing of weakly chordal bipartite graph has a quadratic Gr¨obner basis with respectto a suitable monomial order, which implies that K [ P ] is Koszul.1. Polyominoes and Polyomino ideals
First we recall some definitions and notation from [8]. Given a = ( i, j ) and b =( k, l ) in N we write a ≤ b if i ≤ k and j ≤ l . The set [ a, b ] = { c ∈ N : a ≤ c ≤ b } iscalled an interval . If i < k and j < l , then the elements a and b are called diagonal corners and ( i, l ) and ( k, j ) are called anti-diagonal corners of [ a, b ]. An interval ofthe from C = [ a, a + (1 , cell (with left lower corner a ). The elements(corners) a, a + (0 , , a + (1 , , a + (1 ,
1) of [ a, a + (1 , vertices of C .The sets { a, a + (1 , } , { a, a + (0 , } , { a + (1 , , a + (1 , } and { a + (0 , , a + (1 , } are called the edges of C . We denote the set of edge of C by E ( C ).Let P be a finite collection of cells of N . The the vertex set of P , denoted by V ( P ) is given by V ( P ) = S C ∈P V ( C ). The edge set of P , denoted by E ( P ) is givenby E ( P ) = S C ∈P E ( C ). Let C and D be two cells of P . Then C and D are said tobe connected , if there is a sequence of cells C : C = C , . . . , C m = D of P such that C i ∩ C i +1 is an edge of C i for i = 1 , . . . , m −
1. If in addition, C i = C j for all i = j ,then C is called a path (connecting C and D ). The collection of cells P is called a polyomino if any two cells of P are connected, see Figure 1. Figure 1. polyominoLet P be a polyomino, and let K be a field. We denote by S the polynomial ringover K with variables x ij with ( i, j ) ∈ V ( P ). Following [8] a 2-minor x ij x kl − x il x kj ∈ S is called an inner minor of P if all the cells [( r, s ) , ( r + 1 , s + 1)] with i ≤ r ≤ k − j ≤ s ≤ l − P . In that case the interval [( i, j ) , ( k, l )] is called an inner interval of P . The ideal I P ⊂ S generated by all inner minors of P is calledthe polyomino ideal of P . We also set K [ P ] = S/I P .Let P be a polyomino. Following [4], an interval [ a, b ] with a = ( i, j ) and b = ( k, l )is called a horizontal edge interval of P if j = l and the sets { ( r, j ) , ( r + 1 , j } for r = i, . . . , k − P . If a horizontal edge interval of P is notstrictly contained in any other horizontal edge interval of P , then we call it maximal horizontal edge interval. Similarly one defines vertical edge intervals and maximalvertical edge intervals of P .Let { V , . . . , V m } be the set of maximal vertical edge intervals and { H , . . . , H n } be the set of maximal horizontal edge intervals of P . We denote by G ( P ), theassociated bipartite graph of P with vertex set { v , . . . , v m } F { h , . . . , h n } and the dge set defined as follows E ( G ( P )) = {{ v i , h j } | V i ∩ H j ∈ V ( P ) } . Example 1.1.
The Figure 2 shows a polyomino P with maximal vertical and maxi-mal horizontal edge intervals labelled as { V , . . . , V } and { H , . . . , H } respectively,and Figure 3 shows the associated bipartite graph G ( P ) of P . V V V V V H H H H Figure 2. maximal intervals of P h h h h v v v v v Figure 3. associate bipartite graph of P Let S be the polynomial ring over field K with variables x ij with ( i, j ) ∈ V ( P ).Note that | V p ∩ H q | ≤
1. If V p ∩ H q = { ( i, j ) } , then we may write x ij = x V p ∩ H q ,when required. To each cycle C : v i , h j , v i , h j , . . . , v i r , h j r in G ( P ), we associate abinomial in S given by f C = x V i ∩ H j . . . x V ir ∩ H jr − x V i ∩ H j . . . x V i ∩ H jr .We recall the definition of a cycle in P from [4]. A sequence of vertices C P = a , a , . . . , a m in V ( P ) with a m = a and such that a i = a j for all 1 ≤ i < j ≤ m − cycle in P if the following conditions hold:(i) [ a i , a i +1 ] is a horizontal or vertical edge interval of P for all i = 1 , . . . , m − i = 1 , . . . , m one has: if [ a i , a i +1 ] is a horizontal edge interval of P , then[ a i +1 , a i +2 ] is a vertical edge interval of P and vice versa. Here, a m +1 = a .We set V ( C P ) = { a , . . . , a m } . Given a cycle C P in P , we attach to C P the binomial f C P = ( m − / Y i =1 x a i − − ( m − / Y i =1 x a i oreover, we call a cycle in P is primitive if each maximal interval of P containsat most two vertices of C P .Note that if C : v i , h j , v i , h j , . . . , v i r , h j r defines a cycle in G ( P ) then the se-quence of vertices C P : V i ∩ H j , V i ∩ H j , V i ∩ H j , . . . , V i r ∩ H j r , V i ∩ H j r is aprimitive cycle in P and vice versa. Also, f C = f C P .We set K [ G ( P )] = K [ v p h q | { p, q } ∈ E ( G ( P ))] ⊂ T = K [ v , . . . , v m , h , . . . , h n ].The subalgebra K [ G ( P )] is called the edge ring of G ( P ). Let ϕ : S → T be thesurjective K-algebra homomorphism defined by ϕ ( x ij ) = v p h q , where { ( i, j ) } = V p ∩ H q . We denote by J P , the toric ideal of K [ G ( P )]. It is known from [7], that J P is generated by the binomials associated with cycles in G ( P ).2. Simple polyominoes are prime
Let P be a polyomino and let [ a, b ] an interval with the property that P ⊂ [ a, b ].According to [8], a polyomino P is called simple , if for any cell C not belonging to P there exists a path C = C , C , . . . , C m = D with C i
6∈ P for i = 1 , . . . , m andsuch that D is not a cell of [ a, b ]. For example, the polyomino illustrated in Figure 1is not simple but the one in Figure 4 is simple. It is conjectured in [8] that I P is aprime ideal if P is simple. Figure 4. simple polyominoWe recall from graph theory that a graph is called weakly chordal if every cycle oflength greater than 4 has a chord. In order to prove following lemma, we define somenotation. We call a cycle C P : V i ∩ H j , V i ∩ H j , V i ∩ H j , . . . , V i r ∩ H j r , V i ∩ H j r in P has a self crossing if for some i p ∈ { i , . . . , i r − } and j q ∈ { j , . . . , j s − } there existvertices a = V i p ∩ H i p , b = V i p ∩ H i p +1 , c = V i s ∩ H i s , d = V i s +1 ∩ H i s such that thereexists a vertex e / ∈ { a, b, c, d } such that e ∈ [ a, b ] ∩ [ c, d ]. In this situation e = V i p ∩ H i s .If C is the associated cycle in G ( P ) then It also shows that { v i p , h i s } ∈ E ( G ( P ))and it gives us a chord in C .Let C P : a , a , . . . , a r be a cycle in P which does not have any self crossing. Thenwe call the area bounded by edge intervals [ a i , a i +1 ] and [ a r , a ] for i ∈ { , r − } ,the interior of C P . Moreover, we call a cell C is an interior cell of C P if C belongsto the interior of C P . Lemma 2.1.
Let P be a simple polyomino. Then the graph G ( P ) is weakly chordal.Proof. Let C be a cycle of G ( P ) of length 2 n with n ≥ C P be the associatedprimitive cycle in P . We may assume that C P does not have any self crossing.Otherwise, by following the definition of self crossing, we know that C has a chord. et C : v i , h j , v i , h j , . . . , v i r , h j r and C P : V i ∩ H j , V i ∩ H j , V i ∩ H j , . . . , V i r ∩ H j r , V i ∩ H j r . We may write a = V i ∩ H j , a = V i ∩ H j , a = V i ∩ H j , . . . , a r − = V i r ∩ H j r , a r = V i ∩ H j r . Also, we may assume that a and a belongs to the samemaximal horizontal edge interval. Then a r and a belongs to the same maximalvertical edge interval.First, we show that every interior cell of C P belongs to P . Suppose that we havean interior cell C of C P which does not belong to P . Let J be any interval suchthat P ⊂ J . Then, by using the definition of simple polyomino, we obtain a pathof cells C = C , C , . . . , C t with C i / ∈ P , i = 1 , . . . t and C t is a boundary cell in J . It shows that V ( C ) ∪ V ( C ) ∪ . . . ∪ V ( C t ) intersects atleast one of [ a i , a i +1 ] for i ∈ { , . . . , r − } or [ a r , a ], which is not possible because C P is a cycle in P . Hence C ∈ P . It shows that an interval in interior of C P is an inner interval of P .Let I be the maximal inner interval of C P to which a and a belongs and let b, c the corner vertices of I . We may assume that a and c are the diagonal cornerand a and b are the anti-diagonal corner of I . If b, c ∈ V ( C P ) then primitivity of C implies that C is a cycle of length 4. We may assume that b / ∈ V ( C P ). Let H ′ bethe maximal horizontal edge interval which contains b and c . The maximality of I implies that H ′ ∩ V ( C P ) = ∅ . For example, see Figure 5. Therefore, { v i , h ′ } is achord in C , as desired. a a b c I Figure 5. maximal inner interval (cid:3)
Theorem 2.2.
Let P be a simple polyomino. Then I P = J P .Proof. First we show that I P ⊂ J P . Let f = x ij x kl − x il x kj ∈ I P . Then there existmaximal vertical edge intervals V p and V q and maximal horizontal edge intervals H m and H n of P such that ( i, j ) , ( i, l ) ∈ V p , ( k, j ) , ( k, l ) ∈ V q and ( i, j ) , ( k, j ) ∈ H m ,( i, l ) , ( k, l ) ∈ H n . It gives that φ ( x ij x kl ) = v p h m h n v q = φ ( x il x kj ). This shows f ∈ J P .Next, we show that J P ⊂ I P . It is known from [6] and [7] that toric ideal of weaklychordal bipartite graph is minimally generated by quadratic binomials associatedwith cycles of length 4. It suffices to show that f C ∈ I P where C is a cycle of length4 in G ( P ).Let I be an interval such that P ⊂ I . Let C : h , v , h , v .. Then, C P : a = H ∩ V , a = H ∩ V , a = H ∩ V and a = H ∩ V is the associated cycle in P which also determine an interval in I . Let a and a be the diagonal corners f this interval. We need to show that [ a , a ] is an inner interval in P . Assumethat [ a , a ] is not an inner interval of P , that is, there exist a cell C ∈ [ a , a ]which does not belong to P . Using the fact that P is a simple polyomino, we obtaina path of cells C = C , C , . . . , C r with C i / ∈ P , i = 1 , . . . , r and C r is a cell in I .Then, V ( C ∪ . . . ∪ C r ) intersects atleast one of the maximal intervals H , H , V , V ,say H , which contradicts the fact that H is an interval in P . Hence, [ a , a ] isan inner interval of P and f C ∈ I P . (cid:3) Corollary 2.3.
Let P be a simple polyomino. Then K [ P ] is Koszul and a normalCohen–Macaulay domain.Proof. From [6], we know that J P = I P has squarefree quadratic Gr¨obner basiswith respect to a suitable monomial order. Hence K [ P ] is Koszul. By theoremof Sturmfels [9], one obtains that K [ P ] is normal and then following a theorem ofHochster [1, Theorem 6.3.5], we obtain that K [ P ] is Cohen-Macaulay. (cid:3) A polyomino ideal may be prime even if the polyomino is not simple. The poly-omino ideal attached to the polyomino in Figure 6 is prime. However, the poly-omino ideal attached to the polyomino attached in Figure 7 is not prime. It wouldbe interesting to know a complete characterization of polyominoes whose attachedpolyomino ideals are prime, but it is not easy to answer. However, as a first step, itis already an interesting question to classify polyominoes with only “one hole” suchthat their associated polyomino ideal is prime.
Figure 6. polyomino with prime polyomino ideal
Figure 7. polyomino with “one hole” eferences [1] W. Bruns, J. Herzog, Cohen–Macaulay rings, Cambridge University Press, London, Cam-bridge, New York, (1993)[2] V. Ene, J. Herzog and T. Hibi, Linearly related polyominoes, to appear in J. Alg Comb.[3] V. Ene, A. A. Qureshi and A. Rauf, Regularity of join-meet ideals of distributive lattices,Electron J. Combin. (3) (2013), , 25-28, (1999).[7] H. Ohsugi and T. Hibi, Toric ideals generated by quadratic binomials, Journal of Algebra, , 509-527, (1999).[8] A. A. Qureshi, Ideals generated by 2-minors, collections of cells and stack polyominoes, J.Algebra, , 279303, (2012).[9] B. Sturmfels, Gr¨obner Bases and Convex Polytopes, Amer. Math. Soc., Providence, RI, (1995) Ayesha Asloob Qureshi, Department of Pure and Applied Mathematics, Gradu-ate School of Information Science and Technology, Osaka University, Toyonaka,Osaka 560-0043, Japan
E-mail address : [email protected] Takafumi Shibuta, Institute of Mathematics for Industry, Kyushu University,Fukuoka 819-0395, Japan
E-mail address : [email protected] Akihiro Shikama, Department of Pure and Applied Mathematics, Graduate Schoolof Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
E-mail address : [email protected]@cr.math.sci.osaka-u.ac.jp