Polytopes associated to dihedral groups

In this note we investigate the convex hull of those n ×  n permutation matrices that correspond to symmetries of a regular n -gon. We give the complete facet description. As an application, we show that this yields a Gorenstein polytope, and we determine the Ehrhart h * -vector.

[1]  Gottfried Tinhofer,et al.  Graph isomorphism and theorems of Birkhoff type , 1986, Computing.

[2]  J. Urry.  Complexity , 2006, Interpreting Art.

[3]  Benjamin Nill,et al.  Combinatorial aspects of mirror symmetry , 2007 .

[4]  Shmuel Onn,et al.  Geometry, Complexity, and Combinatorics of Permutation Polytopes , 1993, J. Comb. Theory A.

[5]  David Perkinson,et al.  Some facets of the polytope of even permutation matrices , 2004 .

[6]  Victor V. Batyrev,et al.  Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties , 1993, alg-geom/9310003.

[7]  Christos A. Athanasiadis.  Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley , 2003, math/0312031.

[8]  B. M. Fulk.  MATH , 1992 .

[9]  Richard A. Brualdi,et al.  Convex Polyhedra of Doubly Stochastic Matrices. I. Applications of the Permanent Function , 1977, J. Comb. Theory A.

[10]  Mohamed Omar,et al.  On Volumes of Permutation Polytopes , 2013 .

[11]  M. Bauer,et al.  Triangulations , 1996, Discret. Math..

[12]  Alexander I. Barvinok,et al.  The Distribution of Values in the Quadratic Assignment Problem , 2002, IPCO.

[13]  Robert M. Guralnick,et al.  Permutation polytopes and indecomposable elements in permutation groups , 2006, J. Comb. Theory, Ser. A.

[14]  Tim Römer,et al.  h-Vectors of Gorenstein polytopes , 2007, J. Comb. Theory, Ser. A.

[15]  Igor Pak,et al.  Four Questions on Birkhoff Polytope , 2000 .

[16]  D. Perkinson,et al.  Frobenius polytopes , 2011, 1102.0988.

[17]  Christian Haase,et al.  On permutation polytopes , 2007, 0709.1615.

[18]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[19]  R. Brualdi,et al.  Convex polyhedra of doubly stochastic matrices—IV , 1976 .

[20]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[21]  Michael Joswig,et al.  polymake: a Framework for Analyzing Convex Polytopes , 2000 .

[22]  T. Hibi,et al.  Convex polytopes all of whose reverse lexicographic initial ideals are squarefree , 2001 .

[23]  S. Robins,et al.  Computing the Continuous Discretely , 2015 .

[24]  Benjamin Nill,et al.  Permutation Polytopes of Cyclic Groups , 2011, 1109.0191.

[25]  J. D. Loera,et al.  Triangulations: Structures for Algorithms and Applications , 2010 .

[26]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[27]  Louis J. Billera,et al.  The Combinatorics of Permutation Polytopes , 1994, Formal Power Series and Algebraic Combinatorics.