Polyomino Ideals (Toric ideals and Gröbner...

Polyomino Ideals(Toric ideals and Grobner basis)

Ayesha Asloob QureshiSabancı Universitesi

Ankara-Istanbul Algebraic Geometry and Number Theorymeetings, Spring 2017

March 4, 2017

Toric rings

◮ Let K be a field.

Toric rings

◮ Let K be a field.

◮ S = K [x1, . . . , xn].

Toric rings

◮ Let K be a field.

◮ S = K [x1, . . . , xn].

◮ A = {u1, . . . , um} be a set of m monomials in S .

Toric rings

◮ Let K be a field.

◮ S = K [x1, . . . , xn].

◮ A = {u1, . . . , um} be a set of m monomials in S . The toric

ring of A is the subring K [A] = K [u1, . . . , um] of S .

Toric rings

◮ Let K be a field.

◮ S = K [x1, . . . , xn].

◮ A = {u1, . . . , um} be a set of m monomials in S . The toric

ring of A is the subring K [A] = K [u1, . . . , um] of S .

◮ Let T = K [t1, . . . , tm]

Toric rings

◮ Let K be a field.

◮ S = K [x1, . . . , xn].

◮ A = {u1, . . . , um} be a set of m monomials in S . The toric

ring of A is the subring K [A] = K [u1, . . . , um] of S .

◮ Let T = K [t1, . . . , tm] and define the surjectivehomomorphism

π : T → K [A]

by setting π(ti ) = ui for i = 1, . . . ,m.

Toric rings

◮ Let K be a field.

◮ S = K [x1, . . . , xn].

◮ A = {u1, . . . , um} be a set of m monomials in S . The toric

ring of A is the subring K [A] = K [u1, . . . , um] of S .

◮ Let T = K [t1, . . . , tm] and define the surjectivehomomorphism

π : T → K [A]

by setting π(ti ) = ui for i = 1, . . . ,m.

◮ The toric ideal of A is the kernel of π.

Toric rings

◮ Let K be a field.

◮ S = K [x1, . . . , xn].

◮ A = {u1, . . . , um} be a set of m monomials in S . The toric

ring of A is the subring K [A] = K [u1, . . . , um] of S .

◮ Let T = K [t1, . . . , tm] and define the surjectivehomomorphism

π : T → K [A]

by setting π(ti ) = ui for i = 1, . . . ,m.

◮ The toric ideal of A is the kernel of π.

In other words, the toric ideal of A is the defining ideal of the toricring K [A]

Toric rings

Let A ⊂ S = K [x1, . . . , x8] where

A = {x1x2, x1x4, x2x3, x3x4, x5x6, x5x8, x6x7, x7x8, x4x5}.

Toric rings

Let A ⊂ S = K [x1, . . . , x8] where

A = {x1x2, x1x4, x2x3, x3x4, x5x6, x5x8, x6x7, x7x8, x4x5}.

Here |A| = 9

Toric rings

Let A ⊂ S = K [x1, . . . , x8] where

A = {x1x2, x1x4, x2x3, x3x4, x5x6, x5x8, x6x7, x7x8, x4x5}.

Here |A| = 9 so take T = K [t1, . . . , t9].

Toric rings

Let A ⊂ S = K [x1, . . . , x8] where

A = {x1x2, x1x4, x2x3, x3x4, x5x6, x5x8, x6x7, x7x8, x4x5}.

Here |A| = 9 so take T = K [t1, . . . , t9]. Then

π : T → K [A] = K [x1x2, x1x4, x2x3, x3x4, x5x6, x5x8, x6x7, x7x8, x4x5]

Toric rings

Let A ⊂ S = K [x1, . . . , x8] where

A = {x1x2, x1x4, x2x3, x3x4, x5x6, x5x8, x6x7, x7x8, x4x5}.

Here |A| = 9 so take T = K [t1, . . . , t9]. Then

π : T → K [A] = K [x1x2, x1x4, x2x3, x3x4, x5x6, x5x8, x6x7, x7x8, x4x5]

and the toric ideal ker π = IA of A is (t1t4 − t2t3, t5t8 − t6t7) bysetting

π(t1) = x1x2, . . . , π(t9) = x4x5.

Toric rings

Let A ⊂ S = K [x1, . . . , x8] where

A = {x1x2, x1x4, x2x3, x3x4, x5x6, x5x8, x6x7, x7x8, x4x5}.

Here |A| = 9 so take T = K [t1, . . . , t9]. Then

π : T → K [A] = K [x1x2, x1x4, x2x3, x3x4, x5x6, x5x8, x6x7, x7x8, x4x5]

and the toric ideal ker π = IA of A is (t1t4 − t2t3, t5t8 − t6t7) bysetting

π(t1) = x1x2, . . . , π(t9) = x4x5.

One can see that π(t1t4 − t2t3) = (x1x2)(x3x4)− (x1x4)(x2x3) = 0.

Toric rings

Let A ⊂ S = K [x1, . . . , x8] where

A = {x1x2, x1x4, x2x3, x3x4, x5x6, x5x8, x6x7, x7x8, x4x5}.

Here |A| = 9 so take T = K [t1, . . . , t9]. Then

π : T → K [A] = K [x1x2, x1x4, x2x3, x3x4, x5x6, x5x8, x6x7, x7x8, x4x5]

and the toric ideal ker π = IA of A is (t1t4 − t2t3, t5t8 − t6t7) bysetting

π(t1) = x1x2, . . . , π(t9) = x4x5.

One can see that π(t1t4 − t2t3) = (x1x2)(x3x4)− (x1x4)(x2x3) = 0.

Also, T/IA ∼= K [A].

Binomial prime ideals

◮ Here, a binomial is a polynomial of the form u − v where u

and v are non-constant monomials.

Binomial prime ideals

◮ Here, a binomial is a polynomial of the form u − v where u

and v are non-constant monomials.

◮ A binomial ideal is an ideal generated by binomials.

Binomial prime ideals

◮ Here, a binomial is a polynomial of the form u − v where u

and v are non-constant monomials.

◮ A binomial ideal is an ideal generated by binomials.

◮ It is known that a toric ideal is always a binomial ideal.

Binomial prime ideals

◮ Here, a binomial is a polynomial of the form u − v where u

and v are non-constant monomials.

◮ A binomial ideal is an ideal generated by binomials.

◮ It is known that a toric ideal is always a binomial ideal.

◮ An ideal is a ”binomial prime ideal” if and only if it is a ”toricideal” of some toric ring.

For any integer 1 ≤ t ≤ min{m, n}, the determinantal ideals,which are generated by all t-minors of a m × n matrix X is wellunderstood and discussed in several papers, for example,[Hochster, Eagon (1971)] and [Bruns, Vetter (1988)],

For any integer 1 ≤ t ≤ min{m, n}, the determinantal ideals,which are generated by all t-minors of a m × n matrix X is wellunderstood and discussed in several papers, for example,[Hochster, Eagon (1971)] and [Bruns, Vetter (1988)], and moregenerally the ideals generated by all t-minors of a one and twosided ladders, see for example [Conca (1995)].

For any integer 1 ≤ t ≤ min{m, n}, the determinantal ideals,which are generated by all t-minors of a m × n matrix X is wellunderstood and discussed in several papers, for example,[Hochster, Eagon (1971)] and [Bruns, Vetter (1988)], and moregenerally the ideals generated by all t-minors of a one and twosided ladders, see for example [Conca (1995)].

Motivated by applications in algebraic statistics, ideals generatedby even more general sets of minors have been investigated,including ideals generated by adjacent 2-minors, see [Hosten,Sullivant (2004)], [Hibi, Ohsugi (2006)] and [Herzog, Hibi (2010)],

For any integer 1 ≤ t ≤ min{m, n}, the determinantal ideals,which are generated by all t-minors of a m × n matrix X is wellunderstood and discussed in several papers, for example,[Hochster, Eagon (1971)] and [Bruns, Vetter (1988)], and moregenerally the ideals generated by all t-minors of a one and twosided ladders, see for example [Conca (1995)].

Motivated by applications in algebraic statistics, ideals generatedby even more general sets of minors have been investigated,including ideals generated by adjacent 2-minors, see [Hosten,Sullivant (2004)], [Hibi, Ohsugi (2006)] and [Herzog, Hibi (2010)],and ideals generated by an arbitrary set of 2-minors in a2× n-matrix [Herzog, Hibi, Hreinsdottir, Kahle, Rauh, (2010)].

a11 a12 a13 a14 a15a21 a22 a23 a24 a25a31 a32 a33 a34 a35a41 a42 a43 a44 a45a51 a52 a53 a54 a55a61 a62 a63 a64 a65

.

a11 a12 a13 a14 a15a21 a22 a23 a24 a25a31 a32 a33 a34 a35a41 a42 a43 a44 a45a51 a52 a53 a54 a55a61 a62 a63 a64 a65

.

The 2× 2 minor M(2 5)(2 4) is given by

a22a54 − a24a52

What do we do with these ideals?

Typically one determines for such ideals their Grobner bases,

What do we do with these ideals?

Typically one determines for such ideals their Grobner bases,determines their resolution

What do we do with these ideals?

Typically one determines for such ideals their Grobner bases,determines their resolution and computes their regularity,

What do we do with these ideals?

Typically one determines for such ideals their Grobner bases,determines their resolution and computes their regularity, checkswhether the rings defined by them are normal, Cohen-Macaulay orGorenstein.

Polyominoes

In order to define the polyominoes and polyomino ideals, weintroduce some terminology.

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 .

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 ∈ N2 : a ≤ c ≤ b} is called an interval,

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 ∈ N2 : a ≤ c ≤ b} is called an interval,

2. the interval of the form C = [a, a + (1, 1)] is called a cell.

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 ∈ N2 : a ≤ c ≤ b} is called an interval,

2. the interval of the form C = [a, a + (1, 1)] is called a cell.

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 .

Let C be the following cell.

(i; j) (i + 1; j)

(i; j + 1) (i + 1; j + 1)

Figure: A polyomino

Let C be the following cell.

(i; j) (i + 1; j)

(i; j + 1) (i + 1; j + 1)

Figure: A polyomino

Then V (C ) = {(i , j + 1), (i + 1, j), (i , j + 1), (i + 1, j + 1)}

Let C be the following cell.

(i; j) (i + 1; j)

(i; j + 1) (i + 1; j + 1)

Figure: A polyomino

Then V (C ) = {(i , j + 1), (i + 1, j), (i , j + 1), (i + 1, j + 1)}and E (C ) = {{(i , j), (i + 1, j)}, {(i + 1, j), (i + 1, j +1)}, {(i , j), (i , j + 1)}, {(i , j + 1), (i + 1, j + 1)}}.

Let P be a finite collection of cells of N2. Then two cells C and D

in P are said to be connected, if there is a sequence of cells

C : C1, . . . ,Cm = D

of P such that Ci ∩ Ci+1 is an edge of Ci for i = 1, . . . ,m − 1.The collection of cells P is called a polyomino if any two cells of Pare connected.

Let P be a finite collection of cells of N2. Then two cells C and D

in P are said to be connected, if there is a sequence of cells

C : C1, . . . ,Cm = D

of P such that Ci ∩ Ci+1 is an edge of Ci for i = 1, . . . ,m − 1.The collection of cells P is called a polyomino if any two cells of Pare connected.

Figure: A polyomino

Polyominoes

Polyominoes are, roughly speaking, plane figures obtained byjoining squares of equal size (cells) edge to edge. Their appearanceorigins in recreational mathematics but also has been a subject ofmany combinatorial investigations including tiling problems.

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-minors oftwo-sided ladders. It also includes the meet-join ideals (Hibi ideals)of planar distributive lattices.

Polyomino ideals

Let

◮ P be a polyomino .

Polyomino ideals

Let

◮ P be a polyomino .

◮ V (P) =⋃

C∈PV (C ), the vertex set of P.

Polyomino ideals

Let

◮ P be a polyomino .

◮ V (P) =⋃

C∈PV (C ), the vertex set of P.

◮ K be a field and S = K [xa : a ∈ V (P)].

Polyomino ideals

Let

◮ P be a polyomino .

◮ V (P) =⋃

C∈PV (C ), the vertex set of P.

◮ K be a field and S = K [xa : a ∈ V (P)].

◮ An inner interval I of a polyomino P is an interval with theproperty that all cells inside I belong to P.

Figure: inner interval

Figure: not an inner interval

◮ Let [(i , j), (k , l)] ⊂ N2 be an inner interval of P. Then the

2-minor xijxkl − xilxkj ∈ S is called an inner minor of P.

◮ Let [(i , j), (k , l)] ⊂ N2 be an inner interval of P. Then the

2-minor xijxkl − xilxkj ∈ S is called an inner minor of P.

◮ The ideal IP ⊂ S generated by all inner minors of P is calledthe polyomino ideal of P. We also set K [P] = S/IP .

Given an ideal I generated by an arbitrary set of 2-minors of X , thequestion arises when I is a prime or a radical ideal and what are itsprimary components.

Given an ideal I generated by an arbitrary set of 2-minors of X , thequestion arises when I is a prime or a radical ideal and what are itsprimary components.

It is shown by [Herzog, Hibi, Hreinsdottir, Kahle, Rauh] that I isalways radical if X is a 2× n matrix and the authors give theexplicit primary decomposition of such ideals.

Given an ideal I generated by an arbitrary set of 2-minors of X , thequestion arises when I is a prime or a radical ideal and what are itsprimary components.

It is shown by [Herzog, Hibi, Hreinsdottir, Kahle, Rauh] that I isalways radical if X is a 2× n matrix and the authors give theexplicit primary decomposition of such ideals.

The problem becomes already much more complicated if m, n ≥ 3.Easy examples show that I need not to be radical in general.

Consider the ideal I generated by the 2-minors

ae − bd , bf − ce, dh − eg , ei − fh

of the matrix

a b c

d e f

g h i

.

Consider the ideal I generated by the 2-minors

ae − bd , bf − ce, dh − eg , ei − fh

of the matrix

a b c

d e f

g h i

.

Then cdh − aei ∈√I \ I . So I is not a radical ideal.

Consider the ideal I generated by the 2-minors

ae − bd , bf − ce, dh − eg , ei − fh

of the matrix

a b c

d e f

g h i

.

Then cdh − aei ∈√I \ I . So I is not a radical ideal.

Our main objective is to classify all polyominoes such thatassociated polyomino ideals are prime or radical.

Simple Polyominoes

Let P be a polyomino and let [a, b] an interval with the propertythat P ⊂ [a, b]. A polyomino P is called simple, if for any cell Cnot belonging to P there exists a path C = C1,C2, . . . ,Cm = D

with Ci 6∈ P for i = 1, . . . ,m and such that D is not a cell of [a, b].

Simple Polyominoes

Let P be a polyomino and let [a, b] an interval with the propertythat P ⊂ [a, b]. A polyomino P is called simple, if for any cell Cnot belonging to P there exists a path C = C1,C2, . . . ,Cm = D

with Ci 6∈ P for i = 1, . . . ,m and such that D is not a cell of [a, b].

Roughly speaking, a polyomono is called simple if it has no holes.

Figure: Not simple

Simple Polyominoes

Let P be a polyomino and let [a, b] an interval with the propertythat P ⊂ [a, b]. A polyomino P is called simple, if for any cell Cnot belonging to P there exists a path C = C1,C2, . . . ,Cm = D

with Ci 6∈ P for i = 1, . . . ,m and such that D is not a cell of [a, b].

Roughly speaking, a polyomono is called simple if it has no holes.

Figure: Not simple

For simple polyominoes, in (-, 2012), it was conjectured that IP isa prime ideal.

Convex Polyominoes

As a first step in this direction, we studied the convex polyominoes.

Figure: Not covnex

Convex Polyominoes

TheoremLet P be a convex collection of cells. Then K [P] is a normal

Cohen–Macaulay domain of dimension |V (P)| − |P|.

One of the possible way to show that polyomino ideal (which is abinomial ideal) of a simple polyomino is prime, is to show that it isa toric ideal of a suitable toric ring...

One of the possible way to show that polyomino ideal (which is abinomial ideal) of a simple polyomino is prime, is to show that it isa toric ideal of a suitable toric ring...

Herzog and Madani, (2016) used other techniques to prove thesame result.

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 and thesets {(r , j), (r + 1, j} for r = i , . . . , k − 1 are edges of cells of P. Ifa horizontal edge interval of P is not strictly contained in anyother horizontal edge interval of P, then we call it maximalhorizontal edge interval. Similarly one defines vertical edgeintervals and maximal vertical edge intervals of P.

Following figure shows a polyomino P with maximal vertical andmaximal horizontal edge intervals labelled as {V1, . . . ,V5} and{H1, . . . ,H4} respectively.

V1

V2V3

V4

V5

H1

H2

H3

H4

Figure: maximal intervals of P

Let {V1, . . . ,Vm} be the set of maximal vertical edge intervals and{H1, . . . ,Hn} be the set of maximal horizontal edge intervals of P.We denote by GP , the associated bipartite graph of P with vertexset {v1, . . . , vm}

⊔{h1, . . . , hn} and the edge set defined as follows

E (GP) = {{vi , hj} | Vi ∩ Hj ∈ V (P)}.

V1

V2V3

V4

V5

H1

H2

H3

H4

Figure: maximal intervals of P

h1 h2 h3 h4

v1 v2 v3 v4 v5

Figure: Associated bipartite graph of P

Let T = K [v1, . . . , vm, h1, . . . , hn]. The subalgebra

K [GP ] = K [vphq : {p, q} ∈ E (GP)] ⊂ T

is called the edge ring of GP .

Let T = K [v1, . . . , vm, h1, . . . , hn]. The subalgebra

K [GP ] = K [vphq : {p, q} ∈ E (GP)] ⊂ T

is called the edge ring of GP .

We denote by JP , the toric ideal of K [G (P)].

Let T = K [v1, . . . , vm, h1, . . . , hn]. The subalgebra

K [GP ] = K [vphq : {p, q} ∈ E (GP)] ⊂ T

is called the edge ring of GP .

We denote by JP , the toric ideal of K [G (P)]. It is known from[Hibi, Ohsugi 1999], that JP is generated by the binomialsassociated with cycles in G (P).

1 2 3 4

5 6 7 8

Figure: cycle of length 4

the associated binomial is x16x47 − x46x17.

1 3 5

2 4 6

Figure: chord

1 3 5

2 4 6

Figure: chord

the associated binomial is x12x36x45 − x23x56x14

1 3 5

2 4 6

Figure: chord

the associated binomial is x12x36x45 − x23x56x14= x56(x12x34 − x23x14)− x12(x34x56 − x36x45).

It shows toric ideal of an edge ring is generated in degree 2 if everycycle of length greater than six has a chord.

It shows toric ideal of an edge ring is generated in degree 2 if everycycle of length greater than six has a chord.

Now we come back to the associated bipartite graph of polyomino.

V1

V2V3

V4

V5

H1

H2

H3

H4

Figure: maximal intervals of P

h1 h2 h3 h4

v1 v2 v3 v4 v5

Figure: Associated bipartite graph of P

Lemma. Let P be a simple polyomino. Then the graph GP isweakly chordal.

Lemma. Let P be a simple polyomino. Then the graph GP isweakly chordal.

In this case, JP is generated in degree 2.

Theorem. Let P be a simple polyomino. Then IP = JP .

Theorem. Let P be a simple polyomino. Then IP = JP .

For proof we show that each 2-minor in IP correspond to a 4-cyclein GP and vice versa.

It is known that the universal Grobner basis of toric ideal of abipartite graph consists of binomials associated with cycles.

It is known that the universal Grobner basis of toric ideal of abipartite graph consists of binomials associated with cycles. It hassquarefree initial ideal.

It is known that the universal Grobner basis of toric ideal of abipartite graph consists of binomials associated with cycles. It hassquarefree initial ideal.

It is known from Hibi, Ohsugi that JP has squarefree quadraticGrobner basis with respect to a suitable monomial order.

It is known that the universal Grobner basis of toric ideal of abipartite graph consists of binomials associated with cycles. It hassquarefree initial ideal.

It is known from Hibi, Ohsugi that JP has squarefree quadraticGrobner basis with respect to a suitable monomial order.

Corollary. Let P be a simple polyomino. Then K [P] is Koszul anda normal Cohen–Macaulay domain.

It is known that the universal Grobner basis of toric ideal of abipartite graph consists of binomials associated with cycles. It hassquarefree initial ideal.

It is known from Hibi, Ohsugi that JP has squarefree quadraticGrobner basis with respect to a suitable monomial order.

Corollary. Let P be a simple polyomino. Then K [P] is Koszul anda normal Cohen–Macaulay domain.

Proof Squarefree quadratic Grobner basis implies Koszulness (Bydefinition) and normality (by Strumfels). Then, following atheorem of Hochster, we obtain that K [P] is Cohen–Macaulaydomain.

A polyomino ideal may be prime even if the polyomino is notsimple.

A polyomino ideal may be prime even if the polyomino is notsimple.

Figure: polyomino with prime polyomino ideal

Figure: polyomino with non-prime polyomino ideal

It would be interesting to know a complete characterization ofpolyominoes whose attached polyomino ideals are prime, but it isnot easy to answer. However, as a first step, it is already aninteresting question to classify polyominoes with only “one hole”such that their associated polyomino ideal is prime.

Questions

What is the complete characterization of prime polyominoes.

Questions

What is the complete characterization of prime polyominoes.

What can we say about the ideal generated by inner t-minors of apolyomino?

Questions

What is the complete characterization of prime polyominoes.

What can we say about the ideal generated by inner t-minors of apolyomino?

Does there exist a monomial order such that polyomino ideals havesquarefree Grobner basis? If Yes then it will show that Polyominoideals are always radical.

Questions

What is the complete characterization of prime polyominoes.

What can we say about the ideal generated by inner t-minors of apolyomino?

Does there exist a monomial order such that polyomino ideals havesquarefree Grobner basis? If Yes then it will show that Polyominoideals are always radical.

When convex polyominoes are Gorenstien? (So far, we have acharacterization of Stack polyomino only.)

Thank you!