Growth Series of Cyclotomic and Root Lattices Federico Ardila (San - PowerPoint PPT Presentation

Growth Series of Cyclotomic and Root Lattices Federico Ardila (San Francisco State University) Matthias Beck (San Francisco State University) Serkan Ho¸ sten (San Francisco State University) Julian Pfeifle (Universitat Polit` ecnica de Catalunya) Kim Seashore (University of California Berkeley) Polyhedra, Lattices, Algebra, and Moments IMS Singapore January 2014

Lattices, Monoid Generators, and Growth Series • • • • • • • • • • • • • • • • • • • • ✡❏ ❏ ✡ ✡❏ ❏ ✡ ✑ ◗ ✑ ◗ ◗ • • • ◗ • • • • • • • ✑ ✑ ◗ ◗ ✑ ✑ � � � � ☎☎ ❏✡ ✡ ❏ ✥ ✥ � • • • • • • • • • • • • • • • • • • • • Growth Series of Cyclotomic and Root Lattices Matthias Beck

Cyclotomic Lattices L = Z [ e 2 πi/m ] ∼ = Z ϕ ( m ) M – all m th roots of unity (suitably identified in R ϕ ( m ) ) h m – coordinator polynomial of Z [ e 2 πi/m ] Theorem (Kløve–Parker 1999) The coordinator polynomial of Z [ e 2 πi/p ] , where p is prime, equals h p ( x ) = x p − 1 + x p − 2 + · · · + 1 . Conjectures (Parker 1999) √ m for a palindromic polynomial g of degree ϕ ( √ m ) . m (1) h m ( x ) = g ( x ) (2) h 2 p ( x ) = � p − 3 � x k + x p − 1 − k � � k � p � p − 1 + 2 p − 1 x 2 2 k =0 j =0 j � 1 + x 8 � � x + x 7 � � x 2 + x 6 � � x 3 + x 5 � + 130 x 4 (3) h 15 ( x ) = + 7 + 28 + 79 Growth Series of Cyclotomic and Root Lattices Matthias Beck

Root Lattices Theorem (Conway–Sloane, Bacher–de la Harpe–Venkov 1997) n � n � 2 � x k h A n ( x ) = k k =0 n �� 2 n + 1 � � n �� � x k h B n ( x ) = − 2 k 2 k k k =0 n � 2 n � � x k h C n ( x ) = 2 k k =0 n �� 2 n � � n �� − 2 k ( n − k ) � x k h D n ( x ) = 2 k n − 1 k k =0 Growth Series of Cyclotomic and Root Lattices Matthias Beck

Capturing Growth Series C 4 = P D 2 C 3 • • • • • • • • • • • • • • • • • • • • ✁❅ � ❅ ✁ ❅ � ❅ � ❅ ✁ ❅ � ❅ ✁ ❅ � ❅ • • ✁ • ❅ • • • • • • • ✟✟✟✟✟✟✟✟✟✟ ❅ � ✁ ❅ � ✁ ❅ � ✁ ❅ � ✁ ❅ � • • ✁ • • • • • • • • • • • • • • • • • • Growth Series of Cyclotomic and Root Lattices Matthias Beck

Capturing Growth Series C 4 = P D 2 C 3 • • • • • • • • • • ✁❅ � ❅ ✁ ❅ � ❅ ✁ ❅ � ❅ ✁ ❅ � ❅ ✁ ❅ • • � • ❅ • • • • • • • ✁ ✁❅ ❅ � � ❅ ❅ ✁ ✁ ❅ ❅ � � ❅ ❅ � � ❅ ❅ ✁ ✁ ❅ ❅ � � ❅ ❅ ✁ ✁ ❅ ❅ � � ❅ ❅ • • ✁ ✁ • ❅ • ❅ • • • • • • ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ ✟✟✟✟✟✟✟✟✟✟ ❅ ❅ � � ✁ ✁ ❅ ❅ � � ✁ ✁ ❅ ❅ � � ✁ ✁ ❅ ❅ � � ✁ ✁ ❅ ❅ � � • ✁ • ✁ • • • • • • • • ❅ � ✁ ❅ � ✁ ✁ ❅ � ✁ ❅ � ✁ ❅ � • • • • • • • • • • Growth Series of Cyclotomic and Root Lattices Matthias Beck

Coordinator Polynomials of Root Lattices Theorem (Conway–Sloane, Bacher–de la Harpe–Venkov, 1997) n � n � 2 � x k h A n ( x ) = k k =0 n �� 2 n + 1 � � n �� � x k h B n ( x ) = − 2 k 2 k k k =0 n � 2 n � � x k h C n ( x ) = 2 k k =0 n �� 2 n � � n �� − 2 k ( n − k ) � x k h D n ( x ) = 2 k n − 1 k k =0 Theorem (Ardila–M B–Ho¸ sten–Pfeifle–Seashore) The coordinator polynomi- als of the growth series of root lattices of type A, C, D are the h -polynomials of any unimodular triangulation of the respective polytopes P A n , P C n , P D n . Growth Series of Cyclotomic and Root Lattices Matthias Beck

Cyclotomic Polytopes For two polytopes P ⊂ R d 1 and Q ⊂ R d 2 , each containing the origin in its interior, we define the direct sum P ◦ Q := conv ( P × 0 d 2 , 0 d 1 × Q ) . For a prime p , we define the cyclotomic polytope C p α = C p ◦ C p ◦ · · · ◦ C p . � �� � p α − 1 times Growth Series of Cyclotomic and Root Lattices Matthias Beck

Cyclotomic Polytopes For two polytopes P ⊂ R d 1 and Q ⊂ R d 2 , each containing the origin in its interior, we define the direct sum P ◦ Q := conv ( P × 0 d 2 , 0 d 1 × Q ) . For a prime p , we define the cyclotomic polytope C p α = C p ◦ C p ◦ · · · ◦ C p . � �� � p α − 1 times For two polytopes P = conv ( v 1 , v 2 . . . , v s ) and Q = conv ( w 1 , w 2 , . . . , w t ) we define their tensor product P ⊗ Q := conv ( v i ⊗ w j : 1 ≤ i ≤ s, 1 ≤ j ≤ t ) . Our construction implies for m = m 1 m 2 , where m 1 , m 2 > 1 are relatively prime, that the cyclotomic polytope C m is equal to C m 1 ⊗ C m 2 . For general m , C m = C √ m ◦ C √ m ◦ · · · ◦ C √ m � �� � m √ m times Growth Series of Cyclotomic and Root Lattices Matthias Beck

Coordinator Polynomials of Cyclotomic Lattices Conjectures (Parker 1999) √ m for a palindromic polynomial g of degree ϕ ( √ m ) . m (1) h m ( x ) = g ( x ) (2) h 2 p ( x ) = � p − 3 � x k + x p − 1 − k � � k � p � p − 1 + 2 p − 1 x 2 2 k =0 j =0 j � 1 + x 8 � � x + x 7 � � x 2 + x 6 � � x 3 + x 5 � + 130 x 4 (3) h 15 ( x ) = + 7 + 28 + 79 Theorem (M B–Ho¸ sten) If m is divisible by at most two odd primes then the boundary of the cyclotomic polytope C m admits a unimodular triangulation. Growth Series of Cyclotomic and Root Lattices Matthias Beck

Open Problems Describe the face structure of C m , e.g., in the case m = pq . ◮ Is C m normal for all m ? ◮ S. Sullivant computed that the dual of C 105 is not a lattice polytope, i.e., ◮ C 105 is not reflexive. If we knew that C 105 is normal, a theorem of Hibi would imply that the coordinator polynomial h 105 is not palindromic, and hence that Parker’s Conjecture (1) is not true in general. Growth Series of Cyclotomic and Root Lattices Matthias Beck