Publication Date:
2014-02-26
Description:
\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.
Keywords:
ddc:000
Language:
English
Type:
reportzib
,
doc-type:preprint
Format:
application/postscript
Format:
application/pdf
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