Monomial bases for combinatorial Hopf algebras

Yannic Vargas (University of Potsdam)

15-Oct-2021, 13:00-14:00 (2 years ago)

Abstract: The algebraic structure of a Hopf algebra can often be understood in terms of a poset on the underlying family of combinatorial objects indexing a basis. For example, the Hopf algebra of quasisymmetric functions is generated (as a vector space) by compositions and admits a fundamental (F) basis and a monomial (M) basis, related by the refinement poset on compositions. Analogous bases can be considered for other Hopf algebras, with similar properties to the F basis, e.g. a product described by some notion of shuffle, and a coproduct following some notion of deconcatenation. We give axioms for how these generalised shuffles and deconcatentations should interact with the underlying poset so that a monomial-like basis can be analogously constructed, generalising the approach of Aguiar and Sottile. We also find explicit positive formulas for the multiplication on monomial basis and a cancellation-free and grouping-free formula for the antipode of monomial elements. We apply these results on classical and new Hopf algebras, related by tree-like structures. This is based on "Hopf algebras of parking functions and decorated planar trees", a joint work with Nantel Bergeron, Rafael Gonzalez D'Leon, Amy Pang and Shu Xiao Li.

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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Organizers: Joscha Diehl, Kurusch Ebrahimi-Fard*, Dominique Manchon, Nikolas Tapia*
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