The combinatorics of Hilbert-Poincaré series of matroids

Vecchi, Lorenzo (2023) The combinatorics of Hilbert-Poincaré series of matroids, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in Matematica, 36 Ciclo. DOI 10.48676/unibo/amsdottorato/11116.
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In this thesis we explore the combinatorial properties of several polynomials arising in matroid theory. Our main motivation comes from the problem of computing them in an efficient way and from a collection of conjectures, mainly the real-rootedness and the monotonicity of their coefficients with respect to weak maps. Most of these polynomials can be interpreted as Hilbert--Poincaré series of graded vector spaces associated to a matroid and thus some combinatorial properties can be inferred via combinatorial algebraic geometry (non-negativity, palindromicity, unimodality); one of our goals is also to provide purely combinatorial interpretations of these properties, for example by redefining these polynomials as poset invariants (via the incidence algebra of the lattice of flats); moreover, by exploiting the bases polytopes and the valuativity of these invariants with respect to matroid decompositions, we are able to produce efficient closed formulas for every paving matroid, a class that is conjectured to be predominant among all matroids. One last goal is to extend part of our results to a higher categorical level, by proving analogous results on the original graded vector spaces via abelian categorification or on equivariant versions of these polynomials.

Tipologia del documento
Tesi di dottorato
Vecchi, Lorenzo
Dottorato di ricerca
Settore disciplinare
Settore concorsuale
Parole chiave
Combinatorics, matroids, geometric lattices, integer sequences, real-rooted polynomials, gamma-positivity, poset theory, polytopes, hodge theory, tableaux enumeration, equivariant polynomials, categorification
Data di discussione
30 Novembre 2023

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