Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Gorenstein Fano polytopes arising from two poset
polytopes
Akiyoshi Tsuchiya
Osaka University

Computational Commutative Algebra and Convex Polytopes
1 Aug, 2016, Kyoto University
This talk is joint work with Takayuki Hibi.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Gorenstein Fano polytope
Let P ⊂ Rd be an integral convex polytope of dimension d.
def

P : Fano ⇐⇒ the origin of Rd is a unique integer point
belonging to the interior of P.
P : Gorenstein Fano (reflexive)def⇐⇒ P is Fano and its dual polytope
P ∨ := {x ∈ Rd | hx, yi ≤ 1 for all y ∈ P}
is integral as well.

P : normaldef⇐⇒ for each integer N > 0 and for each a ∈ N P ∩ Zd , there
exist a1 , . . . , aN ∈ P ∩ Zd such that a = a1 + · · · + aN
,where N P = {N α | α ∈ P}.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Example (Gorenstein Fano polytope)

P

P∨

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Example (Non-Gorenstein Fano polytope)(1, 1, 1)(1, −4, 1)(1, 1, − 32 )(−4, 1, 1)(−1, −1, −2)

PP∨

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Example (Non-normal polytope)(1, 1, 2)(0, 1, 0)(1, 1, 1)(1, 0, 0)

P = conv((0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 2))
P ∩ Z3 = {(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 2)}(1, 1, 1) ∈ 2P ∩ Z3

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Unimodular equivalence

P, Q ⊂ Rd : integral convex polytopes of dimension d

P and Q are unimodularly equivalent⇐⇒ There exists a unimodular matrix U ∈ Zd×d
(i.e., det(U ) = ±1) and an integer vector w ∈ Zd
such that Q = fU (P) + w, where fU is the linear
transformation of Rd (i.e., fU (v) = vU for all v ∈ Rd ).
defGorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

ExampleP

Q

P and Q are not unimodularly equivalent.
P and R are unimodularly equivalent.

RGorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

How many?

Theorem (Lagarias-Ziegler, 1991)
There are only finitely many Gorenstein Fano polytopes up tounimodular equivalence in each dimension.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

How many?

Theorem (Lagarias-Ziegler, 1991)
There are only finitely many Gorenstein Fano polytopes up tounimodular equivalence in each dimension.
dimension
1
2
3
4
≥5

] of Gorenstein Fano polytopes
1
16
4,319
473,800,776
openGorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Example (Gorenstein Fano polytopes of dimension 2)

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Faces of Gorenstein Fano polytopes

Theorem (Haase-Melnikov, 2004)
Every integral convex polytope is unimodularly equivalent to a faceof some Gorenstein Fano polytope.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Faces of Gorenstein Fano polytopes

Theorem (Haase-Melnikov, 2004)
Every integral convex polytope is unimodularly equivalent to a faceof some Gorenstein Fano polytope.
Question
Is every normal integral convex polytope unimodularly equivalent
to a face of some normal Gorenstein Fano polytope?

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Partially Ordered SET

Let P = {p1 , . . . , pd } be a partially ordered set.

I ⊂ P : a poset ideal of P⇐⇒ pi ∈ I and pj ∈ P together with pj ≤ pi guarantee pj ∈ I.
def

A ⊂ P : an antichain of P⇐⇒ pi and pj belonging to A with i 6= j are incomparable.
def

σ = i1 i2 · · · id ∈ Sd : a linear extension of P⇐⇒ ia < ib if pia < pib in P .
def

We write J (P ), A(P ) and E(P ) for the set of poset ideals,
antichains and linear extensions of P .

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Example

P :=

p3

p1

p2

J (P ) = f;; fp1g; fp2g; fp1; p2g; fp1; p2; p3gg
A(P ) = f;; fp1g; fp2g; fp3g; fp1; p2gg
E(P ) = f123; 213g

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Order polytopes and Chain polytopes
P
For each subset I ⊂ P , we define ρ(I) = pi ∈I ei ∈ Rd , where
e1 , . . . , ed are the canonical unit coordinate vectors of Rd .
Richard Stanley introduced the order polytope O(P ) and the chain
polytope C(P ) arising from a partially ordered set P .

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Order polytopes and Chain polytopes
P
For each subset I ⊂ P , we define ρ(I) = pi ∈I ei ∈ Rd , where
e1 , . . . , ed are the canonical unit coordinate vectors of Rd .
Richard Stanley introduced the order polytope O(P ) and the chain
polytope C(P ) arising from a partially ordered set P .
Definition
Let P = {p1 , . . . , pd } be a partially ordered set.
O(P ) := conv({ρ(I) | I ∈ J (P )}),
C(P ) := conv({ρ(A) | A ∈ A(P )})Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Example

P :=

p3

p1

p2

J (P ) = f;; fp1g; fp2g; fp1; p2g; fp1; p2; p3gg
A(P ) = f;; fp1g; fp2g; fp3g; fp1; p2gg
E(P ) = f123; 213g

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Example(1; 1; 1)

O(P )

C(P )

Vol(O(P )) = Vol(C(P )) = 31

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Properties of Two Poset Polytopes

In fact,Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Properties of Two Poset Polytopes

In fact,
O(P ) and C(P ) are integral convex polytopes of dimension d.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Properties of Two Poset Polytopes

In fact,
O(P ) and C(P ) are integral convex polytopes of dimension d.
O(P ) and C(P ) are normal.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Properties of Two Poset Polytopes

In fact,
O(P ) and C(P ) are integral convex polytopes of dimension d.
O(P ) and C(P ) are normal.

Vol(O(P )) = Vol(C(P )).

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Properties of Two Poset Polytopes

In fact,
O(P ) and C(P ) are integral convex polytopes of dimension d.
O(P ) and C(P ) are normal.

Vol(O(P )) = Vol(C(P )).

|nO(P ) ∩ Zd | = |nC(P ) ∩ Zd | for any n ≥ 1.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Higher dimensional construction

Let P, Q ⊂ Rd be integral convex polytopes of dimension d.
We setΩ(P, Q) = conv(P × {1} ∪ (−Q) × {−1}) ⊂ Rd+1 ,where −Q = {−α|α ∈ Q}.
Then P and −Q are facets of Ω(P, Q).

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Three types polytope

Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially orderedsets.
We consider the following polytopesΩ(O(P ), O(Q)), Ω(O(P ), C(Q)), Ω(C(P ), C(Q)).
We want to know when these polytopes are normal and Gorenstein
Fano.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

ExampleΩ(O(P ), O(Q))Ω(C(P ), O(Q))
q2

p2
P =

Ω(C(P ), C(Q))

Q=
p1

q1

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

When are these polytopes Gorenstein Fano?

Theorem (Hibi-T)
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially orderedsets.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

When are these polytopes Gorenstein Fano?

Theorem (Hibi-T)
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially orderedsets.
(i) Ω(O(P ), O(Q)) is normal Gorenstein Fano if and only if P
and Q have a common linear extension.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

When are these polytopes Gorenstein Fano?

Theorem (Hibi-T)
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially orderedsets.
(i) Ω(O(P ), O(Q)) is normal Gorenstein Fano if and only if P
and Q have a common linear extension.
(ii) Ω(O(P ), C(Q)) is always normal Gorenstein Fano.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

When are these polytopes Gorenstein Fano?

Theorem (Hibi-T)
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially orderedsets.
(i) Ω(O(P ), O(Q)) is normal Gorenstein Fano if and only if P
and Q have a common linear extension.
(ii) Ω(O(P ), C(Q)) is always normal Gorenstein Fano.

(iii) Ω(C(P ), C(Q)) is always normal Gorenstein Fano.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

ExampleΩ(O(P ), O(Q))Ω(C(P ), O(Q))
q1

p2
P =

Ω(C(P ), C(Q))

Q=
p1

q2

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

How to prove?
We setP ⊂ Rd : An integral convex polytope of dimension d.
S = K[x1 , . . . , xd , t] : polynomial ring.
xα = xα1 1 · · · xαd d for α = (α1 , . . . , αd ) ∈ Zd .
K[P] = K[{xα t : α ∈ P ∩ Zd }] ⊂ S : The toric ring of P.
φ : T = K[{zα : α ∈ P ∩ Zd }] → K[P] (zα 7→ xα t).
IP = ker φ : The toric ideal of P.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

How to prove?
We setP ⊂ Rd : An integral convex polytope of dimension d.
S = K[x1 , . . . , xd , t] : polynomial ring.
xα = xα1 1 · · · xαd d for α = (α1 , . . . , αd ) ∈ Zd .
K[P] = K[{xα t : α ∈ P ∩ Zd }] ⊂ S : The toric ring of P.
φ : T = K[{zα : α ∈ P ∩ Zd }] → K[P] (zα 7→ xα t).
IP = ker φ : The toric ideal of P.
Lemma (Hibi-Matsuda-Ohsugi-Shibata, 2015)dLet
polytope of dimension d. Suppose thatP P ⊂ R be a Fanod+1
Z(α,
1)
=
Z
and there exists a reverse lexicographicdα∈P∩Zorder <rev on T such thatz(0,...,0) is smallest.
in<rev (IP ) is squarefree.
Then P is normal Gorenstein Fano.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propetiesGröbner bases
Set K[OO] = K[{xI }I∈J (P ) ∪ {yJ }J∈J (Q) ∪ {z}].
πOO : K[OO] → K[Ω(O(P ), O(Q))] by setting
πOO (xI ) = tρ(I∪{d+1}) s,
πOO (yJ ) = t−ρ(J∪{d+1}) s,
πOO (z) = s.
Let <OO denote a reverse lexicographic order on K[OO] satisfyingz <OO yJ <OO xI ;xI 0 <OO xI if I 0 ⊂ I;
yJ 0 <OO yJ if J 0 ⊂ J,
and GOO ⊂ K[OO] the set of the following binomials:(i) xI xI 0 − xI∪I 0 xI∩I 0 (I and I 0 are incomparable in J (P ))(ii) yJ yJ 0 − yJ∪J 0 yJ∩J (J and J 0 are incomparable in J (Q))(iii) xI yJ − xI\{pi } yJ\{qi } (pi , qi are maximal elements of I, J)(iv) x∅ y∅ − z 2 ,Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propetiesGröbner bases

Proposition
If P and Q possess a common linear extension, thenΩ(O(P ), O(Q)) is Fano and GOO is a Gröbner basis ofIΩ(O(P ),O(Q)) with respect to <OO .
Similarly, we can show the case of Ω(O(P ), C(Q)) andΩ(C(P ), C(Q)).
Hence the assersion of theorem follows.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Questions

Question
For any (0, 1)-polytope P ⊂ Rd of dimension d, does there exist(0, 1)-polytope Q ⊂ Rd of dimension d such that Ω(P, Q) is a
Gorenstein Fano polytope?

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Questions

Question
For any (0, 1)-polytope P ⊂ Rd of dimension d, does there exist(0, 1)-polytope Q ⊂ Rd of dimension d such that Ω(P, Q) is a
Gorenstein Fano polytope?
Question
For any normal (0, 1)-polytope P ⊂ Rd of dimension d, does there
exist normal (0, 1)-polytope Q ⊂ Rd of dimension d such thatΩ(P, Q) is a normal Gorenstein Fano polytope?

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

ExampleExampleLet P ⊂ R9 be the (0, 1)-polytope of dimension 9 whose vertices
are followings:
e1 + e2 , e2 + e3 , e3 + e4 , e4 + e5 , e1 + e5 , e1 + e6 , e1 + e7 ,e2 + e7 , e2 + e8 , e3 + e8 , e3 + e9 , e4 + e9 , e4 , e5 , e5 + e6 .
Then P is normal. Moreover, Ω(P, P) is Gorenstein Fano, but it is
not normal.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Ehrhart polynomial
Let P ⊂ Rd be an integral convex polytope of dimension d.
We seti(P, n) = |nP ∩ Zd |, n = 1, 2, . . . .
i(P, n) is a polynomial in n of degree d.
We call i(P, n) the Ehrhart polynomial of P.
The following properties are known:Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Ehrhart polynomial
Let P ⊂ Rd be an integral convex polytope of dimension d.
We seti(P, n) = |nP ∩ Zd |, n = 1, 2, . . . .
i(P, n) is a polynomial in n of degree d.
We call i(P, n) the Ehrhart polynomial of P.
The following properties are known:(i) The constant of i(P, n) equals 1;

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Ehrhart polynomial
Let P ⊂ Rd be an integral convex polytope of dimension d.
We seti(P, n) = |nP ∩ Zd |, n = 1, 2, . . . .
i(P, n) is a polynomial in n of degree d.
We call i(P, n) the Ehrhart polynomial of P.
The following properties are known:(i) The constant of i(P, n) equals 1;(ii) The leading coefficient of i(P, n) equals vol(P);Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Ehrhart polynomial
Let P ⊂ Rd be an integral convex polytope of dimension d.
We seti(P, n) = |nP ∩ Zd |, n = 1, 2, . . . .
i(P, n) is a polynomial in n of degree d.
We call i(P, n) the Ehrhart polynomial of P.
The following properties are known:(i) The constant of i(P, n) equals 1;(ii) The leading coefficient of i(P, n) equals vol(P);Remarki(O(P ), n) = i(C(P ), n) for any partially ordered set.
Hence vol(O(P )) = vol(C(P )).

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Hilbert polynomial

In fact,Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Hilbert polynomial

In fact,
Let P ⊂ Rd be an integral convex polytope of dimension d. If
P is normal, then the Ehrhart polynomial of P is equal to theHilbert function of the toric ring K[P].

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Hilbert polynomial

In fact,
Let P ⊂ Rd be an integral convex polytope of dimension d. If
P is normal, then the Ehrhart polynomial of P is equal to theHilbert function of the toric ring K[P].
Let S be a polnomial ring and I ⊂ S be a graded ideal of S.
Let < be a monomial order on S. Then S/I and S/in< (I)have the same Hilbert function.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Toric rings

Proposition
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially orderedsets.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Toric rings

Proposition
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially ordered
sets.
(i) K[OC]/in<OC (IΩ(O(P ),C(Q) ) ∼= K[CC]/in<CC (IΩ(C(P ),C(Q) ).

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Toric rings

Proposition
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially ordered
sets.
(i) K[OC]/in<OC (IΩ(O(P ),C(Q) ) ∼= K[CC]/in<CC (IΩ(C(P ),C(Q) ).
(ii) If P and Q possess a common linear extension, then
K[OO]/in<OO (IΩ(O(P ),O(Q) )∼= K[OC]/in<OC (IΩ(O(P ),C(Q) )∼= K[CC]/in<CC (IΩ(C(P ),C(Q) ).

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Ehrhart polynomials of three types polytopes

Theorem (Hibi-T)
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially orderedsets.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Ehrhart polynomials of three types polytopes

Theorem (Hibi-T)
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially orderedsets.
(i) Ω(O(P ), C(Q)), Ω(C(P ), C(Q)) have the same Ehrhart
polynomial. In particular, these polytopes have the same
volume.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Ehrhart polynomials of three types polytopes

Theorem (Hibi-T)
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially orderedsets.
(i) Ω(O(P ), C(Q)), Ω(C(P ), C(Q)) have the same Ehrhart
polynomial. In particular, these polytopes have the same
volume.
(ii) If P and Q possess a common linear extension, thenΩ(O(P ), O(Q)), Ω(O(P ), C(Q)), Ω(C(P ), C(Q)) have thesame Ehrhart polynomial. In particular, these polytopes havethe same volume.

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

formula of volume

Theorem (Stanley, 1986)
Let P = {p1 , . . . , pd } be a partially ordered set. Then we have
Vol(O(P )) = Vol(C(P )) =]E(P )d!

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

formula of volume
For partially ordered sets P and Q with P ∩ Q = ∅, theordinal sum of P and Q is the partially ordered set P ⊕ Q on
the union P ∪ Q such that s ≤ t in P ⊕ Q if and only if(a) s, t ∈ P and s ≤ t in P , or(b) s, t ∈ Q and s ≤ t in Q, or(c) s ∈ P and t ∈ Q.
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially orderedsets.
For W ⊂ [d] = {1, . . . , d}, We write PW for the partiallyordered set {pi | i ∈ W } such that pi ≤ pj in PW if and only
if pi ≤ pj in P .
For W ⊂ [d], we write ∆W (P, Q) = PW ⊕ QW , whereW = [d] \ W .

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

Example
P :
p1

p3

p2
∆f1;2;3g (P; Q)
p1

p2

p3

q2

∆f1g (P; Q)
q3

p1

Q:
q1

∆f1;2g (P; Q)
q3

q2

q3

p1

p1

p2

p3

∆f2g (P; Q)
q1∆f2;3g (P; Q)
q1∆f1;3g (P; Q)
q2

p2∆f3g (P; Q)
q1

q3

q2

p2

p3

p3
∆; (P; Q)
q1

q2

q3

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

formula of volume
Theorem (Stanley, 1986)
Let P = {p1 , . . . , pd } be a partially ordered set. Then we have
Vol(O(P )) = Vol(C(P )) =]E(P )d!

Gorenstein Fano polytopes
Two Poset Polytopes
Three types polytopes
Combinatorial propeties

formula of volume
Theorem (Stanley, 1986)
Let P = {p1 , . . . , pd } be a partially ordered set. Then we have
Vol(O(P )) = Vol(C(P )) =]E(P )d!

Theorem (Hibi-T)
Let P = {p1 , . . . , pd } and Q = {q1 , . . . , qd } be partially orderedsets, and set P 0 = {pd+1 } ⊕ P and Q0 = {qd+1 } ⊕ Q. If P and Q
have a same linear extension, then we havevol(Ω(O(P ), O(Q))) =X
W ⊂[d+1]]E(∆W (P 0 , Q0 )).
(d + 1)!