growth series of cyclotomic and root lattices
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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


  1. 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

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

  3. 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

  4. 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

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

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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

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