arXiv:1607.04871v1 [math.CO] 17 Jul 2016
SELF DUAL REFLEXIVE SIMPLICES WITH EULERIAN δ-POLYNOMIALS TAKAYUKI HIBI, MCCABE OLSEN AND AKIYOSHI TSUCHIYA Abstract. Given n = 2, 3, . . . , we construct a self dual reflexive simplex Qn of dimension n − 1 such that (i) the normalized volume of Qn is n!, (ii) the δpolynomial of Qn is the Eulerian polynomial of degree n−1, and (iii) Qn possesses a flag and unimodular triangulation.
Self dual reflexive simplices are very rare and the discovery of self dual reflexive simplices has been achieved only by Nill [4] and Tsuchiya [7]. In the present paper, given an integer n ≥ 2, we construct a self dual reflexive simplex Qn of dimension n − 1 with the properties that • the normalized volume of Qn is n!; • the δ-polynomial of Qn is the Eulerian polynomial of degree n − 1; • Qn possesses a flag and unimodular triangulation. The fact that the normalized volume of Qn is n! says that Qn with n ≥ 4 can be unimodularly equivalent to none of the self dual reflexive simplices constructed in [4] and [7]. A lattice polytope is a convex polytope each of whose vertices has integer coordinates. A lattice polytope P ⊂ Rd is called Fano if P is of dimension d and the origin of Rd is a unique integer point belonging to its interior. We say that a Fano polytope P is reflexive if its dual polytope P ∨ ([2, p. 103]) is a lattice polytope. Let Zd×d denote the set of d×d integer matrices. A matrix A ∈ Zd×d is unimodular if det(A) = ±1. Given lattice polytopes P ⊂ Rd and Q ⊂ Rd of dimension d, we say that P and Q are unimodularly equivalent if there exists a unimodular matrix U ∈ Zd×d and a vector w ∈ Zd such that Q = fU (P) + w, where fU is the linear transformation of Rd defined by U, i.e., fU (v) = v U for all v ∈ Rd . A reflexive polytope P ⊂ Rd is called self dual if P and P ∨ are unimodularly equivalent. Let P ⊂ Rd be a lattice polytope of dimension d and δ(P) = (δ0 , δ1 , . . . , δd ) its δ-vector ([2, p. 79]). The δ-polynomial of P is δ(P, λ) = δ0 + δ1 λ + · · · + δd λd . A Fano polytope P ⊂ Rd is reflexive if and only if δi = δd−i for 0 ≤ i ≤ d ([3]). 2010 Mathematics Subject Classification. 13P20, 52B20. Key words and phrases. reflexive polytope, δ-polynomial, Eulerian polynomial. The second author was supported by a 2016 National Science Foundation/Japanese Society for the Promotion of Science East Asia and Pacific Summer Institutes Fellowship. The third author is partially supported by Grant-in-Aid for JSPS Fellows 16J01549. 1
Recall that the descent set of a permutation π = i1 i2 · · · in of [n] = {1, 2, . . . , n} is D(π) = { j : ij > ij+1 } ⊂ [n − 1]. The Eulerian polynomial ([6, p. 33]) of degree P n − 1 is the polynomial An−1 (λ) = π∈Sn λdes(π) , where des(π) = |D(π)|. Now, given an integer n ≥ 2, we introduce the (n − 1)-simplex Qn ⊂ Rn−1 whose vertices are the column vectors of the matrix 1 1−n 0 0 ··· 0 1 1 2−n 0 ··· 0 1 1 1 3 − n · · · 0 . .. . . . . . . . . . . . . . 1 1 1 ··· 1 −1 Theorem thermore, • the • the • Qn
1. The (n − 1)-simplex Qn ⊂ Rn−1 is a self dual reflexive simplex. Furnormalized volume of Qn is n!; δ-polynomial of Qn is the Eulerian polynomial of degree n − 1; possesses a flag and unimodular triangulation.
Proof. The equations of supporting hyperplanes of facets of Qn are as follows: • xn−1 = 1; Pn−1 xi = 1; • − i=1 • −xn−1 − xn−2 − · · · − xi+1 + (n − i)xi = 1, 2 ≤ i ≤ n − 1. It then follows that the dual polytope Q∨n ⊂ Rn−1 is the (n − 1)-simplex whose vertices are the column vectors of the matrix −1 n − 1 0 0 ··· 0 −1 −1 n − 2 0 ··· 0 −1 −1 −1 n − 3 · · · 0 . .. . . . . .. .. . . .. . −1 −1 −1 · · · −1 1 One has Q∨n = −Qn . Thus, in particular, Qn is reflexive and self dual. Furthermore, since the normalized volume of Qn − (1, 1, . . . , 1) is n!, the normalized volume of Qn is n!, as desired. In order to compute the δ-polynomial of Qn and to study triangulations of Qn , we employ the notion of lecture hall polytopes. Given a sequence (s1 , s2 , . . . , sn−1) of (s1 ,s2 ,...,sn−1 ) positive integers, the lecture hall polytope Pn−1 ⊂ Rn−1 is the (n−1)-simplex whose vertices are the column vectors of the matrix 0 sn−1 sn−1 sn−1 · · · sn−1 0 0 sn−2 sn−2 · · · sn−2 0 0 0 sn−3 · · · sn−3 . .. . . . . . . . . . . . . . 0 0 0 ··· 0 s1 2
(n,n−1,...,3,2)
It is routine work to show that Qn is unimodularly equivalent to Pn−1 . In (n,n−1,...,3,2) addition, it is known [5] that the δ-polynomial of Pn−1 coincides with the Eulerian polynomial of degree n − 1. Hence the δ-polynomial of Qn is the Eulerian (1,2,...,n) polynomial of degree n − 1. Since Pn possesses a flag and unimodular tri(1,2,...,n) angulation ([1]) and since Pn is unimodularly equivalent to the pyramid over (n,n−1,...,3,2) Pn−1 , it follows that Qn possesses a flag and unimodular triangulation, as required. References [1] M. Beck, B. Braun, M. K¨ oppe, C. D. Savage and Z. Zafeirakopoulos, Generating functions and triangulations for lecture hall cones, SIAM J. Discrete Math., to appear, arXiv:1508.04619. [2] T. Hibi, “Algebraic Combinatorics on Convex Polytopes,” Carslaw Publications, Glebe, N.S.W., Australia, 1992. [3] T. Hibi, Dual polytopes of rational convex polytopes, Combinatorica 12(1992), 237–s240. [4] B. Nill, Volume and lattice points of reflexive simplicies, Discrete Comput. Geom. 37 (2007), 301–320. [5] C. D. Savage and M. J. Schuster, Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences, J. Combin. Theory Ser. A 119 (2012), 850–870. [6] R. P. Stanley, “Enumerative Combinatorics, Volume I, 2nd ed.,” Cambridge University Press, New York, 2012. [7] A. Tsuchiya, The δ-vectors of reflexive polytopes and of the dual polytopes, Discrete Math. 339 (2016), 2450–2456. Takayuki Hibi, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan E-mail address:
[email protected] McCabe Olsen, Department of Mathematics, University of Kentucky, Lexington, KY 40506–0027, USA E-mail address:
[email protected] Akiyoshi Tsuchiya, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan E-mail address:
[email protected]
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