Combinatorial Hopf Algebras and the Toric Variety of the Permutohedron

29.04.2022, 14:00 (unusual time!)  –  Online Meeting
Arbeitsgruppenseminar Analysis

William Norledge (Penn State, US)

Abstract: We discuss a bimonoid structure on the toric variety of the permutohedron (=permutohedral space) with multiplication, resp. comultiplication, given by embedding, resp. projecting onto, boundary divisors. We show how this bimonoid then induces bimonoid/Hopf algebra structure on data attached to permutohedral space. Many well-known combinatorial gadgets give rise to data on permutohedral space. We shall see that the known Hopf algebra structure enjoyed by these combinatorial gadgets coincides with that induced by permutohedral space. In this way, permutohedral space appears to underlie several important combinatorial Hopf algebras, and also recovers and extends many aspects of Aguiar-Mahajan's interpretation of combinatorial Hopf theory in terms of the braid hyperplane arrangement. Time permitting, we discuss connections to Feynman integrals where this bimonoid structure appears implicitly, and cohomological field theory where the multiplication of permutohedral space appears as an analog of the cyclic operad structure of the moduli space of genus zero stable marked curves.

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