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Polyomino Ideals
Ayesha Asloob Qureshi
Abdus Salam International Centre of Theoretical Physics, Trieste
The 22nd National School on Algebra, 2012Romania, September 1-5, 2014
Ayesha Asloob Qureshi Polyomino Ideals
A. A. Qureshi, Ideals generated by 2-minors, collections ofcells and stack polyominoes, Journal of Algebra, 357, 279–303, (2012).
J. Herzog, A. A. Qureshi, A. Shikama, Grobner basis ofbalanced polyominoes, to appear in Math. Nachr.
Ayesha Asloob Qureshi Polyomino Ideals
Polyominoes
Polyominoes are, roughly speaking, plane figures obtained byjoining squares of equal size (cells) edge to edge. Theirappearance origins in recreational mathematics but also hasbeen a subject of many combinatorial investigations includingtiling problems.
Ayesha Asloob Qureshi Polyomino Ideals
Polyominoes
A connection of polyominoes to commutative algebra has beenestablished first in (-, 2012) by assigning to each polyomino itsideal of inner minors (also called Polyomino ideals).
This class of ideals widely generalizes the ideal of 2-minors of amatrix of indeterminates, and even that of the ideal of 2-minorsof two-sided ladders. It also includes the meet-join ideals (Hibiideals) of planar distributive lattices.
Ayesha Asloob Qureshi Polyomino Ideals
Polyominoes
For any integer 1 t min{m,n}, the ideal generated by allt-minors of X is well understood and discussed in severalpapers, for example, [Hochster, Eagon (1971)] and [Bruns,Vetter (1988)], and more generally the ideals generated by allt-minors of a one and two sided ladders, see for example [Conca(1995)].
Motivated by applications in algebraic statistics, idealsgenerated by even more general sets of minors have beeninvestigated, including ideals generated by adjacent 2-minors,see [Hosten, Sullivant (2004)], [Hibi, Ohsugi (2006)] and[Herzog, Hibi (2010)], and ideals generated by an arbitrary setof 2-minors in a 2⇥ n-matrix [Herzog, Hibi, Hreinsdottir,Kahle, Rauh, (2010)].
Ayesha Asloob Qureshi Polyomino Ideals
Polyominoes
For any integer 1 t min{m,n}, the ideal generated by allt-minors of X is well understood and discussed in severalpapers, for example, [Hochster, Eagon (1971)] and [Bruns,Vetter (1988)], and more generally the ideals generated by allt-minors of a one and two sided ladders, see for example [Conca(1995)].
Motivated by applications in algebraic statistics, idealsgenerated by even more general sets of minors have beeninvestigated, including ideals generated by adjacent 2-minors,see [Hosten, Sullivant (2004)], [Hibi, Ohsugi (2006)] and[Herzog, Hibi (2010)], and ideals generated by an arbitrary setof 2-minors in a 2⇥ n-matrix [Herzog, Hibi, Hreinsdottir,Kahle, Rauh, (2010)].
Ayesha Asloob Qureshi Polyomino Ideals
Polyominoes
Typically one determines for such ideals their Grobner bases,determines their resolution and computes their regularity,checks whether the rings defined by them are normal,Cohen-Macaulay or Gorenstein.
Ayesha Asloob Qureshi Polyomino Ideals
Polyominoes
In order to define the polyominoes and polyomino ideals, weintroduce some terminology.
We denote by < the natural partial order on N2, i.e.(i, j) (k, l) if and only if i k and j l.Let a = (i, j) and b = (k, l) in N2. Then
(1) the set [a, b] = {c 2 N2 : a c b} is called an interval,
(2) the interval of the form C = [a, a+ (1, 1)] is called a cell.(with lower left corner a),
(3) the elements of C are called the vertices of C, and the sets{a, a+ (1, 0)}, {a, a+ (0, 1)}, {a+ (1, 0), a+ (1, 1)} and{a+ (0, 1), a+ (1, 1)} the edges of C.
Ayesha Asloob Qureshi Polyomino Ideals
Polyominoes
In order to define the polyominoes and polyomino ideals, weintroduce some terminology.
We denote by < the natural partial order on N2, i.e.(i, j) (k, l) if and only if i k and j l.
Let a = (i, j) and b = (k, l) in N2. Then
(1) the set [a, b] = {c 2 N2 : a c b} is called an interval,
(2) the interval of the form C = [a, a+ (1, 1)] is called a cell.(with lower left corner a),
(3) the elements of C are called the vertices of C, and the sets{a, a+ (1, 0)}, {a, a+ (0, 1)}, {a+ (1, 0), a+ (1, 1)} and{a+ (0, 1), a+ (1, 1)} the edges of C.
Ayesha Asloob Qureshi Polyomino Ideals
Polyominoes
In order to define the polyominoes and polyomino ideals, weintroduce some terminology.
We denote by < the natural partial order on N2, i.e.(i, j) (k, l) if and only if i k and j l.Let a = (i, j) and b = (k, l) in N2. Then
(1) the set [a, b] = {c 2 N2 : a c b} is called an interval,
(2) the interval of the form C = [a, a+ (1, 1)] is called a cell.(with lower left corner a),
(3) the elements of C are called the vertices of C, and the sets{a, a+ (1, 0)}, {a, a+ (0, 1)}, {a+ (1, 0), a+ (1, 1)} and{a+ (0, 1), a+ (1, 1)} the edges of C.
Ayesha Asloob Qureshi Polyomino Ideals
Polyominoes
Let P be a finite collection of cells of N2, and let C and D betwo cells of P. Then C and D are said to be connected, if thereis a sequence of cells C : C
1
, . . . , Cm = D of P such thatCi \Ci+1
is an edge of Ci for i = 1, . . . ,m� 1. The collection ofcells P is called a polyomino if any two cells of P are connected.
Figure : A polyomino
Ayesha Asloob Qureshi Polyomino Ideals
Polyominoes
Let P be a finite collection of cells of N2, and let C and D betwo cells of P. Then C and D are said to be connected, if thereis a sequence of cells C : C
1
, . . . , Cm = D of P such thatCi \Ci+1
is an edge of Ci for i = 1, . . . ,m� 1. The collection ofcells P is called a polyomino if any two cells of P are connected.
Figure : A polyomino
Ayesha Asloob Qureshi Polyomino Ideals
Polyomino ideals
Let P be a polyomino, and let K be a field. We denote by S
the polynomial over K with variables xij with (i, j) 2 V (P). A2-minor xijxkl � xilxkj 2 S is called an inner minor of P if allthe cells [(r, s), (r + 1, s+ 1)] with i r k � 1 andj s l � 1 belong to P. In that case the interval [(i, j), (k, l)]is called an inner interval of P. The ideal IP ⇢ S generated byall inner minors of P is called the polyomino ideal of P. We alsoset K[P] = S/IP .
Ayesha Asloob Qureshi Polyomino Ideals
Polyomino ideals
Our main objective is to classify all polyominoes such thatassociated polyomino ideals are prime.
Ayesha Asloob Qureshi Polyomino Ideals
Convex Polyominoes
As a first step in this direction, we studied the convex
polyominoes.
Figure : Not covnex
i
Ayesha Asloob Qureshi Polyomino Ideals
Convex Polyominoes
Theorem
Let P be a convex collection of cells. Then K[P] is a normal
Cohen–Macaulay domain of dimension |V (P)|� |P|.
Ayesha Asloob Qureshi Polyomino Ideals
Simple Polyominoes
For simple polyominoes, in (-, 2012), it was conjectured that IPis a prime ideal.
Let P be a polyomino and let [a, b] an interval with theproperty that P ⇢ [a, b]. A polyomino P is called simple, if forany cell C not belonging to P there exists a pathC = C
1
, C
2
, . . . , Cm = D with Ci 62 P for i = 1, . . . ,m and suchthat D is not a cell of [a, b].Roughly speaking, a polyomono is called simple if it has noholes.
Figure : Not simple
Ayesha Asloob Qureshi Polyomino Ideals
Simple Polyominoes
For simple polyominoes, in (-, 2012), it was conjectured that IPis a prime ideal.
Let P be a polyomino and let [a, b] an interval with theproperty that P ⇢ [a, b]. A polyomino P is called simple, if forany cell C not belonging to P there exists a pathC = C
1
, C
2
, . . . , Cm = D with Ci 62 P for i = 1, . . . ,m and suchthat D is not a cell of [a, b].Roughly speaking, a polyomono is called simple if it has noholes.
Figure : Not simple
Ayesha Asloob Qureshi Polyomino Ideals
Admissibe labeling
Let P be a polyomino. An interval [a, b] with a = (i, j) andb = (k, l) is called a horizontal edge interval of P if j = l andthe sets {r, r + 1} for r = i, . . . , k � 1 are edges of cells of P.Similarly one defines vertical edge intervals of P. We call, aninteger value function ↵ : V (P) ! Z is called admissible, if forall maximal horizontal or vertical edge intervals I of P one has
X
a2I↵(a) = 0.
Ayesha Asloob Qureshi Polyomino Ideals
Admissibe labeling
�3 5 �2
0
4
0 �1
�1
2�2
0�2
3
�4
�22
1
0
03
�2
�1
Figure : An admissible labeling
Ayesha Asloob Qureshi Polyomino Ideals
Balanced polyominoes
Given an admissible labeling ↵, we define the binomial
f↵ =Y
a2V (P)
↵(a)>0
x
↵(a)a �
Y
a2V (P)
↵(a)<0
x
�↵(a)a ,
Let JP be the ideal generated by the binomials f↵ where ↵ is anadmissible labeling of P. It is obvious that IP ⇢ JP . We call apolyomino balanced if for any admissible labeling ↵, thebinomial f↵ 2 IP . This is the case if and only if IP = JP .
Ayesha Asloob Qureshi Polyomino Ideals
Balanced polyominoes
Given an admissible labeling ↵, we define the binomial
f↵ =Y
a2V (P)
↵(a)>0
x
↵(a)a �
Y
a2V (P)
↵(a)<0
x
�↵(a)a ,
Let JP be the ideal generated by the binomials f↵ where ↵ is anadmissible labeling of P. It is obvious that IP ⇢ JP . We call apolyomino balanced if for any admissible labeling ↵, thebinomial f↵ 2 IP . This is the case if and only if IP = JP .
Ayesha Asloob Qureshi Polyomino Ideals
Balanced polyominoes
Consider the free abelian group G =L
(i,j)2V (P)
Zeij with basiselements eij . To any cell C = [(i, j), (i+ 1, j + 1)] of P weattach the element bC = eij + ei+1,j+1
� ei+1,j � ei,j+1
in G andlet ⇤ ⇢ G be the lattice spanned by these elements.
Ayesha Asloob Qureshi Polyomino Ideals
Balanced polyominoes
Lemma
The elements bC form a K-basis of ⇤ and hence rankZ ⇤ = |P|.Moreover, ⇤ is saturated. In other words, G/⇤ is torsionfree.
Ayesha Asloob Qureshi Polyomino Ideals
Balanced polyominoes
The lattice ideal I⇤
attached to the lattice ⇤ is the idealgenerated by all binomials
fv =Y
a2V (P)
va>0
x
vaa �
Y
a2V (P)
va<0
x
�vaa
with v 2 ⇤.
Proposition
Let P be a balanced polyomino. Then IP = I
⇤
and has height
|P|.
Ayesha Asloob Qureshi Polyomino Ideals
Balanced polyominoes
The lattice ideal I⇤
attached to the lattice ⇤ is the idealgenerated by all binomials
fv =Y
a2V (P)
va>0
x
vaa �
Y
a2V (P)
va<0
x
�vaa
with v 2 ⇤.
Proposition
Let P be a balanced polyomino. Then IP = I
⇤
and has height
|P|.
Ayesha Asloob Qureshi Polyomino Ideals
Grobner basis of balanced polyominoes
The primitive binomials in P are determined by cycles.
A sequence of vertices C = a
1
, a
2
, . . . , am in V (P) with am = a
1
and such that ai 6= aj for all 1 i < j m� 1 is a called acycle in P if the following conditions hold:
(i) [ai, ai+1
] is a horizonal or vertical edge interval of P for alli = 1, . . . ,m� 1;
(ii) for i = 1, . . . ,m one has: if [ai, ai+1
] is a horizonal intervalof P, then [ai+1
, ai+2
] is a vertical edge interval of P andvice versa. Here, am+1
= a
2
.
Ayesha Asloob Qureshi Polyomino Ideals
Grobner basis of balanced polyominoes
The primitive binomials in P are determined by cycles.
A sequence of vertices C = a
1
, a
2
, . . . , am in V (P) with am = a
1
and such that ai 6= aj for all 1 i < j m� 1 is a called acycle in P if the following conditions hold:
(i) [ai, ai+1
] is a horizonal or vertical edge interval of P for alli = 1, . . . ,m� 1;
(ii) for i = 1, . . . ,m one has: if [ai, ai+1
] is a horizonal intervalof P, then [ai+1
, ai+2
] is a vertical edge interval of P andvice versa. Here, am+1
= a
2
.
Ayesha Asloob Qureshi Polyomino Ideals
Grobner basis of balanced polyominoes
a1
a2
a3
a4
a5
a7
a8
a9
a1
a2
a3
a4
Figure : A cycle and a non-cycle in P
Ayesha Asloob Qureshi Polyomino Ideals
Grobner basis of balanced polyominoes
It follows immediately from the definition of a cycle that m� 1is even. Given a cycle C, we attach to C the binomial
fC =
(m�1)/2Y
i=1
xa2i�1
�(m�1)/2Y
i=1
xa2i
Ayesha Asloob Qureshi Polyomino Ideals
Grobner basis of balanced polyominoes
Theorem
Let P be a balanced polyomino.
(a) Let C be a cycle in P. Then fC 2 IP .
(b) Let f 2 IP be a primitive binomial. Then there exists a
cycle C in P such that each maximal interval of P contains
at most two vertices of C and f = ±fC.
Corollary
Let P be a balanced polyomino. Then IP admits a squarefree
initial ideal for any monomial order.
Ayesha Asloob Qureshi Polyomino Ideals
Grobner basis of balanced polyominoes
Theorem
Let P be a balanced polyomino.
(a) Let C be a cycle in P. Then fC 2 IP .
(b) Let f 2 IP be a primitive binomial. Then there exists a
cycle C in P such that each maximal interval of P contains
at most two vertices of C and f = ±fC.
Corollary
Let P be a balanced polyomino. Then IP admits a squarefree
initial ideal for any monomial order.
Ayesha Asloob Qureshi Polyomino Ideals
Classes of balanced polyominoes
The polyomino P is called tree-like if each subpolyomino of Phas a leaf.
Figure : A tree-like polyomino
Ayesha Asloob Qureshi Polyomino Ideals
Classes of balanced polyominoes
The polyomino P is called tree-like if each subpolyomino of Phas a leaf.
Figure : A tree-like polyomino
Ayesha Asloob Qureshi Polyomino Ideals
Classes of Balanced polyominoes
Figure : Not a tree-like polyomino
Ayesha Asloob Qureshi Polyomino Ideals
Classes of balanced polyominoes
Figure : Column convex but not row convex
Ayesha Asloob Qureshi Polyomino Ideals
Classes of balanced polyominoes
Theorem
Let P be a row or column convex, or a tree-like polyomino.
Then P is balanced and simple.
Ayesha Asloob Qureshi Polyomino Ideals
Classes of Balanced polyominoes
Theorem
A polyomino is simple if and only if it is balanced.
Ayesha Asloob Qureshi Polyomino Ideals
Polyominoes
Very recently those convex polyominoes have been classified in[Ene, Herzog, Hibi (2014) ] whose ideal of inner minors islinearly related or has a linear resolution. For some specialpolyominoes also the regularity of the ideal of inner minors isstudied by [Ene, Rauf, - (2013)].
Ayesha Asloob Qureshi Polyomino Ideals
Questions
What is the complete characterization of prime polyominoes.
Can we find another class of prime polyominoes di↵erent thansimple polyominoes?
What can we say about the ideal generated by inner t-minors ofa polyomino?
Does there exist a monomial order such that polyomino ideals(or any particular class of polyominoe ideals) have quadraticGrobner basis?
When convex polyominoes are Gorenstien?
Ayesha Asloob Qureshi Polyomino Ideals
Questions
What is the complete characterization of prime polyominoes.
Can we find another class of prime polyominoes di↵erent thansimple polyominoes?
What can we say about the ideal generated by inner t-minors ofa polyomino?
Does there exist a monomial order such that polyomino ideals(or any particular class of polyominoe ideals) have quadraticGrobner basis?
When convex polyominoes are Gorenstien?
Ayesha Asloob Qureshi Polyomino Ideals
Questions
What is the complete characterization of prime polyominoes.
Can we find another class of prime polyominoes di↵erent thansimple polyominoes?
What can we say about the ideal generated by inner t-minors ofa polyomino?
Does there exist a monomial order such that polyomino ideals(or any particular class of polyominoe ideals) have quadraticGrobner basis?
When convex polyominoes are Gorenstien?
Ayesha Asloob Qureshi Polyomino Ideals
Questions
What is the complete characterization of prime polyominoes.
Can we find another class of prime polyominoes di↵erent thansimple polyominoes?
What can we say about the ideal generated by inner t-minors ofa polyomino?
Does there exist a monomial order such that polyomino ideals(or any particular class of polyominoe ideals) have quadraticGrobner basis?
When convex polyominoes are Gorenstien?
Ayesha Asloob Qureshi Polyomino Ideals
Questions
What is the complete characterization of prime polyominoes.
Can we find another class of prime polyominoes di↵erent thansimple polyominoes?
What can we say about the ideal generated by inner t-minors ofa polyomino?
Does there exist a monomial order such that polyomino ideals(or any particular class of polyominoe ideals) have quadraticGrobner basis?
When convex polyominoes are Gorenstien?
Ayesha Asloob Qureshi Polyomino Ideals
Thank you!
Ayesha Asloob Qureshi Polyomino Ideals
References
A. Conca, Ladder determinantal rings, J. Pure Appl.Algebra 98, 119–134 (1995)
V. Ene, J. Herzog, T. Hibi, Linearly related polyominoes,arXiv:1403.4349 [math.AC]
J. Herzog, T. Hibi, F. Hreinsdottir, T. Kahle, J. Rauh.Binomial edge ideals and conditional independencestatements, Adv. Appl. Math. 45, 317–333 (2010)
J. Herzog, T. Hibi, Ideals generated by adjacent 2–minors,Journal of Commutative Algebra 4, 525–549 (2012)
J. Herzog, S. Madani, The coordinate ring of a simplepolyomino, arXiv:1408.4275v1, (2014)
Ayesha Asloob Qureshi Polyomino Ideals
References
J. Herzog, A. A. Qureshi, A. Shikama, Grobner basis ofbalanced polyominoes, to appear in Math. Nachr.
M. Hochster and J.A. Eagon, Cohen–Macaulay rings,invariant theory and the generic perfection of determinantalloci. Amer.J.Math. 93, 1020–1058 (1971)
S. Hosten, S. Sullivant, Ideals of adjacent minors, J.Algebra 277 , 615–642 (2004)
H. Ohsugi, T. Hibi, Toric ideals generated by quadraticbinomials. J. Algebra 218, 509–527 (1999)
H. Ohsugi, T. Hibi, Special simplices and gorenstein toricrings. J. Combinatorial Theory Series A 113, (2006)
J. Shapiro, S. Hosten, Primary decomposition of latticebasis ideals. J.Symbolic Computation 29, 625–639 (2000)
Ayesha Asloob Qureshi Polyomino Ideals
References
B. Sturmfels, Grobner Bases and Convex Polytopes, Amer.Math. Soc., Providence, RI, (1995)
B. Sturmfels, D. Eisenbud, Binomial ideals, Duke Math. J.84 , 1–45 (1996)
U. Vetter, W. Bruns, Determinantal rings, Lecture Notes inMathematics, Springer, (1988)
R. Villarreal, Monomial Algebras. Marcel Dekker
A. A. Qureshi, Ideals generated by 2-minors, collections ofcells and stack polyominoes, Journal of Algebra, 357, 279–303, (2012).
Ayesha Asloob Qureshi Polyomino Ideals