Real-rootedness results for triangulation operations inspired by the - PowerPoint PPT Presentation

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Real-rootedness results for triangulation operations inspired by the Tchebyshev polynomials G´ abor Hetyei Department of Mathematics and Statistics University of North Carolina at Charlotte June 27, 2014, Stanley@70. The recent part of the research presented here is joint work with Eran Nevo. G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) The Tchebyshev transform of a poset 1 The Tchebyshev triangulation of a simplicial complex 2 Generalized Tchebyshev triangulations (with Eran Nevo) 3 G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Stanley combination plane G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Stanley combination plane G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) The definition of the Tchebyshev transform G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) The definition of the Tchebyshev transform For each x < y introduce an element ( x , y ). G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) The definition of the Tchebyshev transform For each x < y introduce an element ( x , y ). Set ( x 1 , y 1 ) ≤ ( x 2 , y 2 ) if either y 1 ≤ x 2 or both x 1 = x 2 and y 1 ≤ y 2 hold. G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) The definition of the Tchebyshev transform For each x < y introduce an element ( x , y ). Set ( x 1 , y 1 ) ≤ ( x 2 , y 2 ) if either y 1 ≤ x 2 or both x 1 = x 2 and y 1 ≤ y 2 hold. The resulting poset is the Tchebyshev transform of the original. G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) An example: “The butterfly poset” of rank 3 G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) An example: “The butterfly poset” of rank 3 � 2 � 1 b 1 b 2 a 1 a 2 � 0 � − 1 G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) An example: “The butterfly poset” of rank 3 ( � 1 , � 2) ( � ( a 1 , � 1) − 1 , � 1) ( b 2 , � 1) ( a 2 , � 1) ( b 1 , � ( � 0 , � 1) 1) ( a 1 , b 1) ( � − 1 , b 1) ( a 1 , b 2) ( � − 1 , b 2) ( a 2 , b 1) ( a 2 , b 2) ( � 0 , b 1) ( � 0 , b 2) ( � ( � ( � ( � − 1 , a 1) − 1 , a 2) 0 , a 1) 0 , a 2) ( � − 1 , � 0) G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Properties of the Tchebyshev transform G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Properties of the Tchebyshev transform 1 It preserves the Eulerian property. G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Properties of the Tchebyshev transform 1 It preserves the Eulerian property. 2 The order complex is a triangulation of the original order complex. G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Properties of the Tchebyshev transform 1 It preserves the Eulerian property. 2 The order complex is a triangulation of the original order complex. 3 Takes the Cartesian product of posets into the diamond product of their Tchebyshev transforms (Ehrenborg-Readdy) G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Properties of the Tchebyshev transform 1 It preserves the Eulerian property. 2 The order complex is a triangulation of the original order complex. 3 Takes the Cartesian product of posets into the diamond product of their Tchebyshev transforms (Ehrenborg-Readdy) 4 Induces a Hopf algebra endomorphism on the ring of quasisymmetric functions (Ehrenborg-Readdy) G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Why the name Tchebyshev? G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Why the name Tchebyshev? Definition The F -polynomial of a ( d − 1)-dimensional simplicial complex △ is given by � x − 1 � j d � F ( △ , x ) = f j − 1 2 j =0 G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Why the name Tchebyshev? Definition The F -polynomial of a ( d − 1)-dimensional simplicial complex △ is given by � x − 1 � j d � F ( △ , x ) = f j − 1 2 j =0 The F -polynomial of the order complex of the butterfly poset of rank n + 1 is x n . G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Why the name Tchebyshev? Definition The F -polynomial of a ( d − 1)-dimensional simplicial complex △ is given by � x − 1 � j d � F ( △ , x ) = f j − 1 2 j =0 The F -polynomial of the order complex of the butterfly poset of rank n + 1 is x n . The F -polynomial for its Tchebyshev transform is T n ( x ). G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Why the name Tchebyshev? Definition The F -polynomial of a ( d − 1)-dimensional simplicial complex △ is given by � x − 1 � j d � F ( △ , x ) = f j − 1 2 j =0 The F -polynomial of the order complex of the butterfly poset of rank n + 1 is x n . The F -polynomial for its Tchebyshev transform is T n ( x ). Note to Richard: For an Eulerian poset P , substituting c = x and e = 1 yields into the ce -index F ( △ ( P \ { � 0) , � 1 } , x ). G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Visual definition G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Visual definition ( v 1 , � v 1 1) ( v 1 , v 3 ) ( v 3 , v 4 ) ( v 1 , v 2 ) v 3 v 4 ( v 3 , � ( v 4 , � 1) 1) ( v 2 , v 3 ) ( v 2 , � v 2 1) G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations

Outline The Tchebyshev transform of a poset The Tchebyshev triangulation of a simplicial complex Generalized Tchebyshev triangulations (with Eran Nevo) Visual definition ( v 1 , � v 1 1) ( v 1 , v 3 ) ( v 3 , v 4 ) ( v 1 , v 2 ) v 3 v 4 ( v 3 , � ( v 4 , � 1) 1) ( v 2 , v 3 ) ( v 2 , � v 2 1) In words: pull the midpoint of every edge “in appropriate order”. G´ abor Hetyei (and Eran Nevo) Tchebyshev triangulations