Overview Statistic: PDF-Downloads (blue) and Frontdoor-Views (gray)

Coxeter-associahedra.

Please always quote using this URN: urn:nbn:de:0297-zib-1075
  • \def\KPA{\hbox{\rm KPA}}\def\A{{\rm A}}\def\KPW{\hbox{\rm KPW}}\def\W{{\rm W}}\def\B{{\rm B}} \def\D{{\rm D}} Recently M.~M.~Kapranov [Kap] defined a poset $\KPA_{n-1}$, called the {\it permuto-associahedron}, which is a hybrid between the face poset of the permutahedron and the associahedron. Its faces correspond to the partially parenthesized, ordered, partitions of the set $\{1,2,\ldots,n\}$, with a natural partial order. Kapranov showed that $\KPA_{n-1}$ is the face poset of a CW-ball, and explored its connection with a category-theoretic result of MacLane, Drinfeld's work on the Knizhnik-Zamolodchikov equations, and a certain moduli space of curves. He also asked the question of whether this CW-ball can be realized as a convex polytope. We show that this permuto-associahedron corresponds to the type $\A_{n-1}$ in a family of convex polytopes $\KPW$ associated to each of the classical Coxeter groups, $\W = \A_{n-1}, \B_n, \D_n$. The embedding of these polytopes relies on the secondary polytope construction of the associahedron due to Gel'fand, Kapranov, and Zelevinsky. Our proofs yield integral coordinates, with all vertices on a sphere, and include a complete description of the facet-defining inequalities. Also we show that for each $\W$, the dual polytope $\KPW^*$ is a refinement (as a CW-complex) of the Coxeter complex associated to $\W$, and a coarsening of the barycentric subdivision of the Coxeter complex. In the case $\W=\A_{n-1}$, this gives an elementary proof of Kapranov's original sphericity result.

Download full text files

Export metadata

Additional Services

Share in Twitter Search Google Scholar Statistics - number of accesses to the document
Metadaten
Author:Victor Reiner, Günter M. Ziegler
Document Type:ZIB-Report
Date of first Publication:1993/05/02
Series (Serial Number):ZIB-Report (SC-93-11)
ZIB-Reportnumber:SC-93-11
Published in:Appeared in: Mathematika 41 (1994) pp. 364-393
Accept ✔
Diese Webseite verwendet technisch erforderliche Session-Cookies. Durch die weitere Nutzung der Webseite stimmen Sie diesem zu. Unsere Datenschutzerklärung finden Sie hier.