Abstract It is known that toric ring of a simple polyomino is ring homomorphic to a edge ring of a weakly chordal bipartite graph. In this paper we identify the attached toric rings of nonsimple polyominoes which are of the form &quot;rectangle minus rectangle&quot;. Mathematics Subject Classification: 13C05, 05E4

Akihiro Shikama

CiteSeerX

International Journal of Algebra, Vol. 9, 2015, no. 4, 195 - 201
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ija.2015.5312
Toric Rings of Nonsimple Polyominoes
Akihiro Shikama
Department of Pure and Applied Mathematics
Graduate School of Information Science and Technology
Osaka University, Toyonaka, Osaka 560-0043, Japan
Copyright c© 2015 Akihiro Shikama. This article is distributed under the Creative Com-
mons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Abstract
It is known that toric ring of a simple polyomino is ring homomorphic to a edge
ring of a weakly chordal bipartite graph. In this paper we identify the attached toric
rings of nonsimple polyominoes which are of the form “rectangle minus rectangle”.
Mathematics Subject Classification: 13C05, 05E40
Keywords: polyominoes, toric ideals
Introduction
Polyominoes are two dimensional objects which are originally rooted in recre-
ational mathematics and combinatorics. They have been widely discussed in
connection with tiling problems of the plane. Typically, a polyomino is plane
figure obtained by joining squares of equal sizes, which are known as cells. In
connection with commutative algebra, polyominoes are first discussed in [5] by
assigning each polyomino the ideal of inner 2-minors or the polyomino ideal.
The study of ideal of t-minors of an m×n matrix is a classical subject in com-
mutative algebra. The class of polyomino ideal widely generalizes the class
of ideals of 2-minors of m × n matrix as well ass 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 IP , the polyomino
ideal attatched to P , in a suitable polynomial ring over K. The residue class
ring defined by IP denoted by K[P ]. It is natural to investigate the algebraic
properties of IP depending on shape of P . The classes of polyominoes whose
polyomino ideal is prime has been discussed in many papers. In [5] it is shown
196 Akihiro Shikama
that polyomino ideal of convex polyominoes are prime. In [3] they introduced
the class of balanced polyominoes and proved that polyomino ideal of balanced
polyominoes are prime. Later in [4] it is shown that simple polyominoes are
balanced. In [6], independently from [4], it is proved that polyomino ideal of
simple polyominoes are prime by identifying them with toric rings of edge rings
of graphs. Very recently, in [2] it is shown that polyomino ideal of polyomino
of the shape “rectangle minus convex” is prime by using localization argument
on the attached rings of those polyominoses.
In this paper, we identify the toric rings of polyomino ideals for the special
class of nonsimple polyominoes, “rectangle minus rectangle”.
1. Toric rings of nonsimple polyominoes
First we recall some definitions and notation from [5]. Given a = (i, j) and
b = (k, l) in N2 we write a ≤ b if i ≤ k and j ≤ l. The set [a, b] = {c ∈
N2 : a ≤ c ≤ b} is called 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 of the from C = [a, a+ (1, 1)] is called a cell (with
left lower corner a). The elements (corners) a, a + (0, 1), a + (1, 0), a + (1, 1)
of [a, a + (1, 1)] are called the vertices of C. The sets {a, a + (1, 0)}, {a, a +
(0, 1)}, {a+ (1, 0), a+ (1, 1)} and {a+ (0, 1), a+ (1, 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 N2. The the vertex set of P , denoted
by V (P) is given by V (P) =
⋃
C∈P V (C). For each vertex v in V (P), we write
v = (v1, v2) where v1 is the first and v2 is the second coodinate of v. The edge
set of P , denoted by E(P) is given by E(P) =
⋃
C∈P E(C). Let C and D be
two cells of P . Then C and D are said to be connected, if there is a sequence
of cells C : C = C1, . . . , Cm = D of P such that Ci ∩ Ci+1 is an edge of Ci for
i = 1, . . . ,m − 1. If in addition, Ci 6= Cj for all i 6= 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. polyomino
Let P be a polyomino, and let K be a field. We denote by S the polynomial
ring over K with variables xv with v ∈ V (P). We write for v = (i, j), we may
Toric rings of nonsimple polyominoes 197
write xv = xij if needed. Following [5] a 2-minor xijxkl − xilxkj ∈ S is called
an inner minor of P if all the cells [(r, s), (r + 1, s + 1)] with i ≤ r ≤ k − 1
and j ≤ 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 by all inner minors of P
is called the polyomino ideal of P . We also set K[P ] = S/IP .
For each interval [a, b] we regard the polyomino P(a,b) in the obvious way.
We give Theorem 1.2 that identifies a toric ring for some specified cases of
nonsimple polyominoes which is obtained by subtracting a rectangle from a
bigger rectangle.
Hereafter let P be a polyomino with P = P(a,b)\P(a′b′) with (a < a′ < b′ < b).
We define a map µ : V (P)→ Z as follows. For v = (v1, v2) in P ,
µ(v) =

1 (for a1 ≤ v1 ≤ a′1, a2 ≤ v2 ≤ a′2)
1 (for b′1 ≤ v1 ≤ b1, b′2 ≤ v2 ≤ b2)
2 (for a1 ≤ v1 ≤ a′1, b′2 ≤ v2 ≤ b2)
2 (for b′1 ≤ v1 ≤ b1, a2 ≤ v2 ≤ a′2)
v1 − a′1 + 2 (for a′1 < v1 < b′1, a2 ≤ v2 ≤ b′2)
v2 − a′2 + (b′1 − a′1) + 1
(for b′1 ≤ v1 ≤ b1, a′2 < v2 < b′2)
b′1 − v1 + (b′1 − a′1) + (b′2 − a′2)
(for a′1 < v1 < b
′
1, b
′
2 ≤ v2 ≤ b2)
b′2 − v2 + 2(b′1 − a′1) + (b′2 − a′2)− 1
(for a1 ≤ v1 ≤ a′1, a′2 < v2 < b′2).
We call µ(v) the labelling of v.
Example 1.1. for a = (1, 1) b = (7, 5), a′ = (2, 2) and b′ = (5, 4), the labelling
of vertices are given as follows.
2
1
1 1
1 2 2 2
2 2 23
3
4
4
5 5 5
1 1 1
1 1 1
6
6
7
7
2
2
2
8 8
Figure 2. labelling of P
We put 1 or 2 for each corners and 3, 4, . . . for the other vertices anti-
clockwisely.
Let T = K[rv1sv2tµ(v) | v ∈ V (P)] be the subring of a polynomial ring
K[ra1 , . . . rb1 , sa2 , . . . sb2 , t1 . . . tM ] where M = max{µ(v) | v ∈ V (P)}. We de-
fine the surjective ring homomorphism ϕ : K[P ] → T by setting ϕ(xv) =
198 Akihiro Shikama
rv1sv2tµ(v). For the sake of convenience, we write ϕ(v) = rv1sv2tµ(v) = rvsvtv in
case it does not make readers confused. We call the toric ideal JP = kerϕ
Theorem 1.2. Let P = P(a,b) \ P(a′b′) with (a < a′ < b′ < b). Then IP = JP .
Proof. First it is easy to see that IP ⊂ kerϕ. We show that every quadratic
binomials belong to JP is a generator of IP , namely an inner 2-minor in P and
every binomials of higher degree belongs to IP .
First, suppose that xv1xv2−xv3xv4 is a quadratic binomial belongs to JP with
xv1xv2 6= xv3xv4 . Then either (rv1 , rv2 , sv1sv2) = (rv3 , rv4 , sv4sv3) or (rv1 , rv2 , sv1sv2) =
(rv4 , rv3 , sv3sv4) holds. Thus, the quadratic binomial xv1xv2 − xv3xv4 is associ-
ated with an interval. Now we show that the interval is an inner interval in P .
Assume that the interval spanned by xv1xv2 −xv3xv4 has v1 and v2 as diagonal
corners. Also, we may assume thatrv1 = rv3 and sv1 = sv4 hold.
Suppose [v1, v2] is not an inner interval in P . Then P[a′,b′] ⊂ P[v1,v2] can not
be happened because otherwise, from our definition of µ, one has t21 = tv1tv2 6=
tv3tv4 = t
2
2. Therefore we may assume that one of [v1, v3], [v1, v4], [v3, v2] or
[v4, v2] is not an inner interval in P . Then from our definition of labelling we
can not have tv1tv2 = tv3tv4 . Hence every degree 2 binomial in JP is a generator
of IP .
Now we show that each binomial of higher degree belongs to IP . Let f =
f (+) − f (−) ∈ kerϕ be a homogeneous binomial. Let V+ = v1 . . . vs be the set
of vertices such that xvi appear in f
(+) and V− = u1 . . . xut the vertices appear
in f (−).
We may assume that V+ ∩ V− = ∅, otherwise, say if v1 = u1 holds, we have
f = xv1(f
(+)/xv1−f (−)/xu1) and we can check f ′ = f (+)/xv1−f (−)/xu1 instead
of f to complete the proof.
Now we show that if we have v ∈ V+ ∪ V− such that µ(v) /∈ {1, 2}, then
we are done in this case. Assume that we have a vertex u1 in V− such that
u11 ≤ a′1 and µ(v) 6= 1, 2. Since f belongs to the kernel, we have 2 vertices, say
v1 and v2 in V+ such that sv1tv1|ϕ(xu1) and rv1 |ϕ(xu1). Let c be the rest corner
of the interval spanned by v1, v2 and u1. Our rule of labelling show that the
interval is an inner interval in P See Figure 3 to see the situation. We obtain
u1
v1
v2
c
Figure 3
Toric rings of nonsimple polyominoes 199
f = f (+) − f (−)
= xv1xv2
f (+)
xv1xv2
− xc
f (−)
xc
= (xv1xv2 − xu1xc)
f (+)
xv1xv2
+ xu1xc
f (+)
xv1xv2
− xc
f (−)
xc
= (xv1xv2 − xu1xc)
f (+)
xv1xv2
+ xc
(
xu1
f (+)
xv1xv2
− f
(−)
xc
)
Since xv1xv2 − xu1xc is an inner minor of P , we have xu1(f (+)/xv1xv2) −
(f (−)/xc) ∈ JP , we can apply induction on the degree of f to complete the
proof.
Now we assume every vertex in V(+) and V(−) are labelled 1 or 2. For
u1 ∈ V(−), we can find a vertex v1 and v2 in V+ where ru1 = rv1 and su1 = sv2 .
If tu1 = tv1 or tu1 = tv2 , then using the same formula as above, we can apply
induction. Now we assume that tu1 = t1 and tv1 = tv2 = t2. By using similar
argument, one has u2 ∈ V− such that su2 = sv1 . Since ϕ(f (+)) = ϕ(f (−)) holds,
we have a vertex u2 ∈ V− such that tu2 = t1. See Figure 4 for this situation.
Then the interval spanned by u1 and u3 is an inner minor of P . Let c, d be the
u1
v1
v2
u2
u3 d
c
e
Figure 4
other corners of this interval. Then,
f = f (+) − f (−)
= f (+) − xu1xu3
f (−)
xu1xu3
= f (+) − xcxd
f (−)
xu1xu3
− (xu1xu3 − xcxd)
f (−)
xu1xu3
Notice that one of tc = t1 or td = t1 holds. Assume td = t1. Then the interval
spanned by u2 and d has v1 as a corner vertex. Suppose that e is the rest
200 Akihiro Shikama
corner vertex of this interval. Then, we have
f =f (+) − xcxd
f (−)
xu1xu3
− (xu1xu3 − xcxd)
f (−)
xu1xu3
=xv1
f (+)
xv1
− xcxdxu2
f (−)
xu1xu2xu3
− (xu1xu3 − xcxd)
f (−)
xu1xu3
=xv1
f (+)
xv1
− xv1xcxe
f (−)
xu1xu2xu3
− (xdxu2 − xv1xe)
(
xc
f (−)
xu1xu2xu3
)
− (xu1xu3 − xcxd)
f (−)
xu1xu3
=xv1
(
f (+)
xv1
− xcxe
f (−)
xu1xu2xu3
)
− (xdxu2 − xv1xe)
(
xc
f (−)
xu1xu2xu3
)
− (xu1xu3 − xcxd)
f (−)
xu1xu3
.
Since (xdxu2 − xv1xe) and (xu1xu3 − xcxd) are inner minors of P and since
degree of f (+)/xv1 − xcxef (−)/xu1xu2xu3 is less than degree of f , we can apply
induction on degree of f to complete the proof. 
Example 1.3. A toric ring of the polyomino given in Example 1.1 is identified
by Theorem 1.2 as follows.
S/IP ∼= K

r1s5t2 r2s5t2 r3s5t7 r4s5t6 r5s5t1 r6s5t1 r7s5t1
r1s4t2 r2s4t2 r3s4t7 r4s4t6 r5s4t1 r6s4t1 r7s4t1
r1s3t8 r2s3t8 r5s3t5 r6s3t5 r7s3t5
r1s2t1 r2s2t1 r3s2t3 r4s2t4 r5s2t2 r6s3t2 r7s2t2
r1s1t1 r2s1t1 r3s1t3 r4s1t4 r5s1t2 r6s2t2 r7s1t2
 .
In this paper we identify toric rings of polyominoes which are of the form
“rectangle minus rectangle”. In general it is interesting but not so easy to iden-
tify toric ring for a given binomial prime ideal. The wider class of polyominoes
which can be discussed about its toric ring is “rectangle minus convex”, be-
cause it is proved in [2] that their polyomino ideals are prime.
References
[1] V. Ene, J. Herzog and T. Hibi, Linearly related polyominoes, J. Alg Comb. 41, 949-968,
(2014). http://dx.doi.org/10.1007/s10801-014-0560-3
[2] T. Hibi and A. A. Qureshi, Nonsimple polyominoes and prime ideals, arXiv:
1502.03669[math.AC].
[3] J. Herzog, A. A. Qureshi and A. Shikama, Gröbner basis of balanced polyominoes,
Math. Nachr. 288, 775-783, (2015). http://dx.doi.org/10.1002/mana.201400153
[4] J. Herzog and S. S. Madani, The coordinate ring of a simple polyomino,
arxiv:1408.4275v1
[5] A. A. Qureshi, Ideals generated by 2-minors, collections of cells and stack polyominoes,
J. Algebra, 357, 279-303, (2012). http://dx.doi.org/10.1016/j.jalgebra.2012.01.032
Toric rings of nonsimple polyominoes 201
[6] A. A. Qureshi, T. Shibuta and A. Shikama, Simple polyominoes are prime, arXiv:
1501.15107[math.AC]
Received: March 17, 2015; Published: May 15, 2015


Toric Rings of Nonsimple Polyominoes

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=285EE1C52E27BBCB9320A09A9303F2AB?doi=10.1.1.1049.2242&rep=rep1&type=pdf

Toric Rings of Nonsimple Polyominoes

Abstract

Similar works

Full text

Available Versions

CiteSeerX