Lattice simplices of maximal dimension
with a given degree
It was proved by Nill that for any lattice simplex of dimension with degree which is not a lattice pyramid, the inequality holds. In this paper, we give a complete characterization of lattice simplices satisfying the equality, i.e., the lattice simplices of dimension with degree which are not lattice pyramids. It turns out that such simplices arise from binary simplex codes. As an application of this characterization, we show that such simplices are counterexamples for the conjecture known as “Cayley conjecture”. Moreover, by modifying Nill’s inequaitly slightly, we also see the sharper bound , where for . We also observe that any lattice simplex attaining this sharper bound always comes from a binary code.
Keywords: lattice simplex, degree, Cayley decomposition, binary linear code.
The author is partially supported by JSPS Grant-in-Aid for Young Scientists (B) 17K14177.
We say that a convex polytope is a lattice polytope if all of its vertices belong to the standard lattice . For two lattice polytopes , we say that and are unimodularly equivalent if there exist and such that . One of the main topics of the study on lattice polytopes is to give a classification of lattice polytopes up to unimodular equivalence.
For a lattice polytope of dimension , we consider the generating function , called the Ehrhart series. It is well known that Ehrhart series becomes a rational function which is of the form
where is a polynomial in with integer coefficients. The polynomial appearing in the numerator of Ehrhart series is called the -polynomial of . Let denote the degree of the -polynomial of . It is known that
where denotes the interior of . In particular, . Moreover, coincides with the volume of , so using the notation is natural. We refer the reader to  for more detailed information on Ehrhart series or -polynomials.
For a lattice polytope , let
This new lattice polytope is said to be a lattice pyramid over . It is not so hard to see that ([4, Theorem 2.4]). In particular, .
1.2. Main Results
The following is one of the most interesting open problems in the theory of lattice polytopes:
Given a nonnegative integer , classify all lattice polytopes with degree which are not lattice pyramids over lower-dimensional ones up to unimodular equivalence.
Let denote the th unit vector of and its origin. Then any lattice polytope of dimension with degree 0 is unimodularly equivalent to the -folded lattice pyramids over one lattice point (-dimensional lattice polytope), i.e., . Moreover, Batyrev and Nill completely solve Problem 1.1 for the case ().
On the other hand, Nill proved the following:
Theorem 1.2 ([10, Theorem 7]).
Let and be nonnegative integers. For a lattice polytope of dimension having at most vertices with , if is not a lattice pyramid over a lower-dimensional one, then holds. In particular, when is a simplex (i.e. ), we have .
In this paper, we give a complete characterization of lattice simplices of maximal dimension for a given degree , i.e., of dimension with degree , which are not lattice pyramids up to unimodular equivalence.
Theorem 1.3 (Main Result 1).
Given a positive integer , let be a lattice simplex of dimension with degree which is not a lattice pyramid. Then for some . Moreover, is uniquely determined by up to unimodular equivalence and arises from the -dimensional binary simplex code. More precisely, is unimodularly equivalent to .
We will explain the binary simplex codes and clarify the lattice simplex arising from a binary simplex code in Section 3.
Furthermore, by modifying Theorem 1.2, we obtain the following (see Proposition 2.3): For a lattice simplex of dimension with degree which is not a lattice pyramid, we have the inequality , where for . Note that holds in general and the bound is sharp. As the second main result of this paper, we will observe that lattice simplices of dimension with degree which are not lattice pyramids satisfying have the special property as follows.
Theorem 1.4 (Main Result 2).
Given a positive integer , let be a lattice simplex of dimension with degree which is not a lattice pyramid. Then arises from a binary code. More precisely, we have .
We will explain what is in Section 2.
1.3. Cayley Conjecture
Recently, Cayley polytopes or Cayley decompositions of lattice polytopes are well studied and play an essential role for the study of lattice polytopes ([1, 5, 6, 7, 9]). Let us recall the notion of Cayley polytopes and Cayley decompositions.
Let be lattice polytopes. The Cayley sum of them is the convex hull of in , where denote the unit vectors of . The lattice polytope of the form is called a Cayley polytope. A Cayley decomposition of a lattice polytope is a choice of unimodular equivalence classes of which is a Cayley sum of some lattice polytopes.
By definition, we see that a lattice polytope is a Cayley sum of lattice polytopes if and only if is mapped onto a unimodualr simplex of dimension by a projection . Note that algebro-geometric interpretation of Cayley polytopes is also known by .
The following, known as Cayley conjecture, is one central problem which concerns a Cayley decomposition of lattice polytopes.
Conjecture 1.6 ([5, Conjecture 1.2]).
Let be a lattice polytope of dimension with degree . If , then is a Cayley polytope of at least lattice polytopes.
Several partial answers for this conjecture are known. For example, this is true for smooth polytopes () or Gorenstein polytopes (). Moreover, by using the invariant satisfying , called -codegree, it is proved in  that if , then is a Cayley polytope of at least lattice polytopes. (Refer to [6, Theorem 3.4] for the precise statement.) Moreover, a weak version of this conjecture is solved in [7, Theorem 1.2], i.e., a bound for the number of Cayley summands is given by a quadratic of the degree.
In Section 5, we will claim that this conjecture does not hold in general. More precisely, we provide an example of a lattice simplex of dimension with degree such that but is a Cayley polytope into less than lattice polytopes. Actually, those counterexamples come from the lattice simplices appearing in Theorem 1.3. Moreover, we will suggest a modification of Conjecture 1.6. We remark that the examples given in this paper leave the possibility open that still implies that is a Cayley polytope of at least two lattice polytopes.
This paper is organized as follows: In Section 2, we introduce a finite abelian group associated with a lattice simplex, which plays an essential role for the classification of unimodular equivalence classes for lattice simplices, and we prepare some lemmas. In Section 3, we introduce a binary simplex code and the finite abelian group arising from it and discuss its properties. In Section 4, we prove Theorem 1.3 and Theorem 1.4. In Section 5, we supply a counterexample for Conjecture 1.6.
The author would like to express a lot of thanks to Johannes Hofscheier and Benjamin Nill for many advices and helpful comments on the main results and Cayley conjecture. The author also would like to thank to Kenji Kashiwabara for many fruitful and intriguing discussions. The essential ideas of the proofs of the main results come from the discussions with him.
2. Finite abelian groups associated with lattice simplices
In this section, we introduce the finite abelian group associated with a lattice simplex and discuss some properties on a lattice simplex in terms of this group.
Let be a lattice simplex of dimension with its vertices . We introduce
equipped with its addition defined by for and . We can see that is a finite abelian group.
Let denote the set of unimodular equivalence classes of lattice simplices of dimension with a fixed vertex order and let denote the set of finite abelian subgroups of satisfying that the sum of all entries of each element in is an integer.
Actually, the correspondence
provides a bijection ([2, Theorem 2.3]). In particular, a unimodular equivalence class of lattice simplices is uniquely determined by the finite abelian group up to permutation of coordinates.
We can discuss , , and whether is a lattice pyramid in terms of . We fix some notation: For , where each is taken with ,
let , where for each ;
Then we have
Consult, e.g., [4, Corollary 3.11]. In particular,
Lemma 2.1 (cf. [2, Proposition 2.5]).
Let be a lattice simplex of dimension . Then is not a lattice pyramid if and only if there is such that for each .
We consider a finite abelian subgroup of (not necessarily the sum of the entries is an integer), i.e., is more general than . We use the same notation and for as above. We also use the notation
Notice that a lattice simplex of dimension is not a lattice pyramid if and only if by Lemma 2.1.
Lemma 2.2 (cf. [10, Lemma 11]).
Let be a finite abelian subgroup of and let . Then for each .
For each , let denote the inverse of . Then
For a positive integer , let
Let be a finite abelian group of with and let with . Assume that . Then the following assertions hold.
One has (cf. [10, Theorem 10]).
If , then for some .
Let . Then there exist some elements such that holds. Conversely, if there exist with , then .
Although most parts of the statements can be obtained just by modifying the proof of [10, Theorem 10] slightly, we give a precise proof for the completeness.
(a) First, we show the inequalities . Let with maximal. We choose some elements successively in a “greedy” manner such that is maximal, where for and . What we may prove is the inequality
In fact, since , we obtain
from (2.1) and , and we also obtain
We prove (2.1) by induction on the number of possible elements . The case is clear. Let and suppose that the assertion is true for . Set . Then we have . Since is chosen with maximal, we obtain
On the other hand, by considering , we also have . By the maximality of again, we obtain
(b) Assume . Then one has by (a). Thus it directly follows from (2.3) that must be a power of .
(c) Assume . Then the equalities of the inequalities in (2.2) are satisfied. In particular, one has for each . This implies that the number of possible elements should be at least , otherwise . ∎
We also see the following observation:
Given an even number and an integer , let be a finite abelian subgroup of with . Let and and assume that
Then and for each .
Let and as in the statement. Note that .
First, consider . Since , it follows that
Next, consider . Since and holds for each by (2.6), we have for each . Similarly, by considering , we obtain
3. Binary simplex codes and the associated lattice simplices
In this section, we introduce some elements of linear codes, especially, binary codes. We associate the finite abelian group of (i.e., the lattice simplex of dimension ) with a binary simplex code . Binary simplex codes and the associated lattice simplices will play the important role in this paper.
Linear subspaces of the vector space over a finite field are called a linear code. Let be a finite field of prime order . We call a linear code binary if . We set the map defined by for . Then, for a linear code , can be regarded as a finite abelian subgroup of .
We will often use the following notation in the remaining parts.
For a positive integer , we consider all the points in -dimensional projective space over . There are points in . For each of those points in , we associate the column vector and we denote by the -matrix whose columns are those vectors. For example, and
Let denote the matrix . More precisely, is the matrix obtained by replacing in into . For example, and
An -dimensional binary simplex code is a binary linear code generated by the row vectors of . Note that the binary simplex code is equivalently a dual code of Hamming code.
Simplex codes play the central role in the paper  for the classification of lattice simplices of dimension which are not lattice pyramids whose -polynomials are of the form , where , and .
Throughout this paper, we will only treat a binary simplex code, while simplex codes are considered for any finite field.
Given , let denote the abelian group arising from -dimensional binary simplex code, i.e., is the finite abelian group generated by the row vectors of . Since it is known that for any , we see that the sum of all entries of each element in is an integer. Thus, we can associate a lattice simplex from . Let denote a lattice simplex corresponding to . Namely, .
We note some properties on , or equivalently, .
The -polynomial of is of the form with . In particular, .
Moreover, we observe the following lemma which we will use in the proof of Lemma 4.1:
Given , let be a finite abelian subgroup of with . Let and let be the matrix whose row vectors are those . Assume that the support matrix of is equal to . Then .
Here, the support vector of a (row or column) vector means the -vector such that if and if , and the support matrix of a matrix means the -matrix whose row (or column) vectors consist of the support vectors of the row (or column) vectors of .
Proof of Lemma 3.2.
Since the support matrix of is equal to , we see that and for each . Hence, by Lemma 2.4. Therefore, one obtains , as required. ∎
Throughout this section, let be a lattice simplex of dimension with degree satisfying which is not a lattice pyramid. We will use the following notation.
Let be the matrix whose row vectors are .
Let be the finite abelian subgroup generated by .
4.1. Key Lemma
The following lemma will be the key for the proofs of our main results.
Lemma 4.1 (Key Lemma).
The matrix contains as a submatrix. In particular, when , coincides with .
For each , let
First, we claim that for any . Let for some , i.e., a singleton. Suppose , i.e., . Then we have , a contradiction to Proposition 2.3 (c). Hence, . Let , where . We set
for . Then we see that
Thus, . Hence, by the above discussion, we have . Moreover, we also see that
Hence, we conclude that for any non-empty . This implies that all non-zero -vectors of appear in as support vectors of its column vectors. In other words, contains a certain submatrix whose support matrix is equal to . Therefore, by Lemma 3.2, we obtain that , as required. ∎
Proof of Theorem 1.3.
(The first step): Let be the same as (4.1) for . Then we saw that . Moreover, we can also see by definition that for any with . Hence, one has and each should be a singleton.
(The second step): We will claim that if , then
Let and fix . Then, by the first step, there exists a unique such that .
For , let
Then . (See the proof of Lemma 4.1.) Let us consider the finite abelian subgroup generated by , denoted by . Then it follows that . Hence, applying the same discussion as in the proof of Lemma 4.1, we obtain that is equal to . In particular, one sees that and .
On the other hand, for each , there exists a unique such that by the first step. For , where , we see that if and only if is odd. Since there are exactly such ’s in , we obtain that is