Lattice simplices of maximal dimension with a given degree Akihiro - PowerPoint PPT Presentation

Lattice simplices of maximal dimension with a given degree Akihiro Higashitani (Kyoto Sangyo University) Einstein Workshop on Lattice Polytopes at Berlin 12–16 Dec. 2016 1

Contents 1. Introduction to Cayley Conjecture and Nill’s bound 2. Correspondence between lattice simplices and finite abelian groups (Johannes’ talk) 3. The case d + 1 = 4 s − 1 and Cayley Conjecture 4. The case d + 1 = f (2 s ) and Cayley Conjecture (j.w.w. K. Kashiwabara) 2

1.1. Introduction to Cayley Conjecture Let P ⊂ R d be a lattice polytope , i.e., P is a convex polytope whose vertices are the points in Z d . P ◦ : the interior of P dim P = d • codeg( P ) := min { k : kP ◦ ∩ Z d � = ∅} • deg( P ) := d + 1 − codeg( P ) Example P (0,0,1) codeg( P ) = 3 deg( P ) = 3 + 1 − 3 = 1 (0,1,0) (1,0,0) (1,1,0) 3

Why do we say deg( P ) degree of P ? Remark For a lattice polytope P ⊂ R d , we consider the n ≥ 0 | nP ∩ Z d | t n . Then this becomes a rational Ehrhart series � function of the form h ∗ P ( t ) | nP ∩ Z d | t n = � (1 − t ) d +1 , n ≥ 0 where h ∗ P ( t ) is a polynomial in t . We say that h ∗ P ( t ) is the h ∗ -polynomial of P . (the degree of h ∗ ( t )) = deg( P ) . 4

For a lattice polytope P ⊂ R d , a lattice pyramid over P is defined by Pyr( P ) := conv( { ( α, 0) ∈ R d +1 : α ∈ P }∪{ (0 , . . . , 0 , 1) } ) ⊂ R d +1 . Then dim(Pyr( P )) = dim P + 1. In particular, those are not unimod. equiv., however... d+1 R Remark Pyr(P) We have h ∗ P ( t ) = h ∗ Pyr( P ) ( t ), in R d particular, deg( P ) = deg(Pyr( P )) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ . P 5

Motivation We want to know Cayley structure of lattice polytopes. ✓ ✏ Cayley Polytope • P 0 , P 1 , . . . , P ℓ ⊂ R d : lattice polytopes P 0 ∗ P 1 ∗· · ·∗ P ℓ := conv(( P 0 × 0 ) ∪ ( P 1 × e 1 ) ∪· · ·∪ ( P ℓ × e ℓ )) ⊂ R d + ℓ We say P 0 ∗ · · · ∗ P ℓ is a Cayley polytope . R d + ℓ , • For a lattice polytope P ⊂ when there exist P 0 , P 1 , . . . , P ℓ ⊂ R d s.t. P ∼ = P 0 ∗ · · · ∗ P ℓ , we say P 0 ∗ · · · ∗ P ℓ is a Cayley decomposition of P . • For a lattice polytope P , let C ( P ) := max( { ℓ +1 : ∃ P 0 , . . . , ∃ P ℓ s.t. P ∼ = P 0 ∗· · ·∗ P ℓ } ) . ✒ ✑ 6

✓ ✏ (Strong) Cayley Conjecture (Dickenstein–Nill ’12) Let P be a lattice polytope of dimension d with degree s . ⇒ C ( P ) ≥ d + 1 − 2 s . d > 2 s = ✒ ✑ ✓ ✏ (Weak) Cayley Conjecture Let P be a lattice polytope of dimension d with degree s . ⇒ C ( P ) ≥ 2, d > 2 s = namely, P can be just decomposed into at least two polytopes. ✒ ✑ Strong Cayley conjecture is true if • P : smooth (Dickenstein–Nill ’10) • P : Gorenstein (DiRocco–Haase–Nill–Paffenholz ’13) • some class of (0 , 1)-polytopes? (work in progress) 7

Theorem (Haase–Nill–Payne ’09) Let P be a lattice polytope of dimension d with degree s . ⇒ C ( P ) ≥ d + 1 − ( s 2 + 19 s − 4) / 2 d > ( s 2 + 19 s − 4) / 2 = Remark ∃ counterexample (appear later) for strong Cayley Conjecture The existence of counterexample for weak Cayley Conjecture might be still open. − → I want to know C ( P ) in order to give its “sharp” bound. I expect the bound of C ( P ) can be given like d + 1 − (linear of s ) . 8

1.2. (modified) Nill’s bound On the other hand, the following theorem is known: ∞ � m � � For m ∈ Z > 0 , let f ( m ) = . 2 ℓ ℓ =0 Theorem (Nill 2008, H. 2016) P : lattice simplex of dimension d with degree s ⇒ d + 1 ≤ f (2 s ) ≤ 4 s − 1 P is NOT a lattice pyramid = Moreover, f (2 s ) is sharp but f (2 s ) < 4 s − 1 in general. (Explain later more precisely.) 9

Thus, it is natural to study the following problem: ✓ ✏ Problem Give a complete characterization of lattice simplices of d + 1 = f (2 s ) . dimension d with degree s satisfying ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ ✒ ✑ Remark • A complete characterization of lattice polytopes of degree 1 which are not lattice pyramids was given by Batyrev–Nill (2007). simplices of degree 2 • A complete characterization of lattice ✿✿✿✿✿✿✿✿ which are not lattice pyramids was given by H.–Hofscheier (2016+). 10

2. Correspondence between lattice simplices and finite abelian groups ✓ ✏ We will review the correspondence between unimodular equvalence classes of lattice simplices and finite abelian groups . ✒ ✑ ∆ ⊂ R d : lattice simplex of dimension d v 0 , v 1 , . . . , v d ∈ Z d : vertices of ∆ � � d d ( x 0 , x 1 , . . . , x d ) ∈ ( R / Z ) d +1 : x i v i ∈ Z d and � � Λ ∆ = x i ∈ Z i =0 i =0 11

Example Λ ∆ = (0,2) v △ { 0 , (1 / 2 , 1 / 2 , 0) , (1 / 2 , 0 , 1 / 2) , (0 , 1 / 2 , 1 / 2) } = 2 ∼ � (1 / 2 , 1 / 2 , 0) , (1 / 2 , 0 , 1 / 2) � = ( Z / 2 Z ) 2 (0,0) (2,0) v v 0 1 Λ ∆ forms a finite abelian group . In this way, from a lattice simplex ∆, we can construct a finite abelian subgroup Λ ∆ of ( R / Z ) d +1 . 12

FACTS (the volume of ∆) · d ! = (the order of Λ ∆ ) �� d � i =0 x i ∈ Z ≥ 0 : ( x 0 , . . . , x d ) ∈ Λ ∆ , 0 ≤ x i < 1 deg(∆) = max ∆ is NOT a lattice pyramid ⇐ ⇒ 0 ≤ ∀ i ≤ d , ∃ x ∈ Λ ∆ s.t. x i � = 0 On the other hand, from a finite abelian subgroup Λ ⊂ ( R / Z ) d +1 s.t. the sum of entries of each element in Λ is an integer. · · · · · · ( ∗ ) we can construct a lattice simplex of dim d . 13

✓ ✏ Correspondence (Batyrev–Hofscheier ’13) { lattice simplices of dim d } / ( unimod. equiv. ) 1:1 ← → { fin. abel. subgroups Λ ⊂ ( R / Z ) d +1 with ( ∗ ) } / (permute of coord.) ✒ ✑ Remark → Λ ∆ ⊂ ( R / Z ) d +1 d + 1 (dimension of ∆) ← s (degree of ∆) ← → maximum of entry sums of Λ ∆ NOT a lattice pyramid ← → 0 ≤ ∀ i ≤ d , ∃ x ∈ Λ ∆ s.t. x i � = 0 14

3. The case d + 1 = 4 s − 1 Recall For a lattice simplex ∆ of dim d with deg s , we have d + 1 ≤ f (2 s ) ≤ 4 s − 1. ✓ ✏ Our Goal Give a complete characterization of lattice simplices of d + 1 = f (2 s ) . dimension d with degree s satisfying ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ ✒ ✑ First, we consider the case d + 1 = 4 s − 1 , which automatically implies d + 1 = f (2 s ). 15

Binary Simplex Codes C ⊂ ( Z / 2 Z ) d +1 : binary simplex code ⇐ ⇒ Binary simplex code is a binary code generated by the row vectors of the matrix H ( d + 1) { column vectors of H ( d + 1) } = { T ( a 1 , . . . , a d +1 ) � = 0 : a i ∈ { 0 , 1 }} Example   1 1 1 0 1 0 0    1 1 0   H (2) = H (3) = 1 1 0 1 0 1 0    1 0 1   1 0 1 1 0 0 1 binary simplex code ⇐ ⇒ a dual code of Hamming code 16

We can identify a binary code C ⊂ F d +1 as a finite abelian 2 subgroup Λ ⊂ { 0 , 1 / 2 } d +1 . We may replace 0 ∈ F 2 ← → 0 ∈ { 0 , 1 / 2 } ⊂ R / Z and 1 ∈ F 2 ← → 1 / 2 ∈ { 0 , 1 / 2 } ⊂ R / Z . Example (0,2) △ v 2 Λ ∆ = � (1 / 2 , 1 / 2 , 0) , (1 / 2 , 0 , 1 / 2) � comes from H (2) . (0,0) (2,0) v v 0 1 By Batyrev–Nill (2007), we know that this triangle is a unique lattice simplex of dim 2 with deg 1 s.t. d + 1 = 4 s − 1. 17

Theorem (H. ’16) ∆ : lattice simplex of dim d with deg s satisfying d + 1 = 4 s − 1 Then s = 2 r for some r ∈ Z ≥ 0 and Λ ∆ comes from binary simplex codes . r ∈ Z ≥ 0 Proposition (H. ’16) ∆( r ) : lattice simplex of dim d with deg s = 2 r s.t. d + 1 = 4 s − 1 Then C (∆( r )) = 4 s − 1 = d + 1 . 3 3 ✓ ✏ We can see that ∆( r ) becomes a counterexample for Strong Cayley Conjecture if r ≥ 1. ✒ ✑ 18

The following looks strange and unnatural... ✓ ✏ Question (Modified Strong Cayley Conjecture?) Let P be a lattice polytope of dimension d with degree s . d > 8 s − 2 ⇒ C ( P ) ≥ d + 1 − 8 s − 2 = ? 3 3 ✒ ✑ ∆( r ) satisfies this conjecture for any r ≥ 0. Remark ∆( r ) is the “most extremal” among the simplices ∆ of dim d with deg s s.t. d + 1 = f (2 s ). − → MSSC is always true for any simplices? 19

4. The case d + 1 = f (2 s ) (j.w.w. K. Kashiwabara) Recall For a lattice simplex ∆ of dim d with deg s , we have d + 1 ≤ f (2 s ) ≤ 4 s − 1. We want to give a complete characterization of lattice simplices of dim d with deg s satisfying d + 1 = f (2 s ) and compute C (∆) (for checking MSCC). ∞ � m � � By the way . . . . . . what is f ( m ) = ?? 2 ℓ ℓ =0 Example f (2) = 2 + 1 , f (3) = 3 + 1 = 4 , f (4) = 4 + 2 + 1 , f (5) = 5 + 2 + 1 , f (6) = 6 + 3 + 1 , f (7) = 7 + 3 + 1 , f (8) = 8 + 4 + 2 + 1 , f (9) = 9 + 4 + 2 + 1 . . . . . . 20

Proposition f ( m ) = 2 m − ( ♯ of 1’s for the binary expansion of m ) In particular, f ( m ) = 2 m − 1 if and only if m is a power of 2. Theorem (again) (H. ’16) ∆ : lattice simplex of dim d with deg s satisfying d + 1 = 4 s − 1 s = 2 r for some r ∈ Z ≥ 0 (obvious from above Prop) and Λ ∆ Then ✿✿✿✿✿✿ comes from binary simplex codes . In particular, Λ ∆ ⊂ { 0 , 1 / 2 } d +1 . Theorem Let ∆ be a lattice simplex ∆ of dim d with deg s s.t. d + 1 = f (2 s ). ⇒ Λ ∆ ⊂ { 0 , 1 / 2 } d +1 = 21