Convex Geometries via Hopf Monoids: Combinatorial Invariants, Reciprocity, and Supersolvability
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908425668001792 |
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| author | Ma, Yichen |
| author_facet | Ma, Yichen |
| contents | We study the Hopf monoid of convex geometries, which contains partial orders as a Hopf submonoid, and investigate the combinatorial invariants arising from canonical characters. Each invariant consists of a pair: a polynomial and a more general quasisymmetric function. We give combinatorial descriptions of the polynomial invariants and prove combinatorial reciprocity theorems for the Edelman-Jamison and Billera-Hsiao-Provan polynomials, which generalize the order and enriched order polynomials, respectively, within a unified framework. For the quasisymmetric invariants, we show that their coefficients enumerate faces of certain simplicial complexes, including subcomplexes of the Coxeter complex and a simplicial sphere structure introduced by Billera, Hsiao, and Provan. We also examine the associated $ab$- and $cd$-indices. We establish an equivalent condition for convex geometries to be supersolvable and use this result to give a geometric interpretation of the $ab$- and $cd$-index coefficients for this class of convex geometries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00380 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convex Geometries via Hopf Monoids: Combinatorial Invariants, Reciprocity, and Supersolvability Ma, Yichen Combinatorics We study the Hopf monoid of convex geometries, which contains partial orders as a Hopf submonoid, and investigate the combinatorial invariants arising from canonical characters. Each invariant consists of a pair: a polynomial and a more general quasisymmetric function. We give combinatorial descriptions of the polynomial invariants and prove combinatorial reciprocity theorems for the Edelman-Jamison and Billera-Hsiao-Provan polynomials, which generalize the order and enriched order polynomials, respectively, within a unified framework. For the quasisymmetric invariants, we show that their coefficients enumerate faces of certain simplicial complexes, including subcomplexes of the Coxeter complex and a simplicial sphere structure introduced by Billera, Hsiao, and Provan. We also examine the associated $ab$- and $cd$-indices. We establish an equivalent condition for convex geometries to be supersolvable and use this result to give a geometric interpretation of the $ab$- and $cd$-index coefficients for this class of convex geometries. |
| title | Convex Geometries via Hopf Monoids: Combinatorial Invariants, Reciprocity, and Supersolvability |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.00380 |