Cyclic polylopes through the lense of iterated integrals
Keywords:
Piecewise linear paths, Shuffle algebra, Group action, Signed volume, Invariants, Positive MatricesAbstract
The volume of a cyclic polytope can be obtained by forming an it-
erated integral, known as the path signature, along a suitable piecewise
linear path running through its edges. Different choices of such a path are
related by the action of a subgroup of the combinatorial automorphisms
of the polytope. Motivated by this observation, we look for other polyno-
mials in the vertices of a cyclic polytope that arise as path signatures and
are invariant under the subgroup action. We prove that there are infinitely
many such invariants which are algebraically independent in the shuffle
algebra.
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