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Honeycomb tessellations and canonical bases for permutohedral blades. (arXiv:1810.03246v1 [math.CO])
来源于:arXiv
This paper studies two families of piecewise constant functions which are
determined by the $(n-2)$-skeleta of collections of honeycomb tessellations of
$\mathbb{R}^{n-1}$ with standard permutohedra. The union of the codimension $1$
cones obtained by extending the facets which are incident to a vertex of such a
tessellation is called a blade.
We prove ring-theoretically that such a honeycomb, with 1-skeleton built from
a cyclic sequence of segments in the root directions $e_i-e_{i+1}$, decomposes
locally as a Minkowski sum of isometrically embedded components of hexagonal
honeycombs: tripods and one-dimensional subspaces. For each triangulation of a
cyclically oriented polygon there exists such a factorization. This
consequently gives resolution to an issue proposed and developed by A. Ocneanu,
to develop a structure theory for an object he introduced during his study of
higher Lie theory: permutohedral blades.
We introduce a certain canonical basis for a vector space spanned by
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