$f$-vectors of balanced simplicial complexes, flag spheres, and geometric Lefschetz decompositions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913541637799936 |
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| author | Park, Soohyun |
| author_facet | Park, Soohyun |
| contents | We show that there are $f$-vectors of balanced simplicial complexes giving a source of simplicial complexes exhibiting a Boolean decomposition similar to a geometric Lefschetz decomposition. The objects we are working with are $h$-vectors of flag spheres and balanced simplicial complexes whose $f$-vectors are equal to them. This builds on work of Nevo--Petersen--Tenner on a conjecture of Nevo--Petersen that the gamma vector of an odd-dimensional flag sphere is the $f$-vector of a balanced simplicial complex (which was shown for barycentric subdivisions by Nevo--Petersen--Tenner). We can connect our decomposition to positivity questions on reciprocal/palindromic polynomials associated to flag spheres and geometric questions motivating them. In addition, we note that the degrees in the Lefschetz-like decomposition are not halved unlike the usual $h$-vector setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_08139 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $f$-vectors of balanced simplicial complexes, flag spheres, and geometric Lefschetz decompositions Park, Soohyun Combinatorics Algebraic Geometry Geometric Topology We show that there are $f$-vectors of balanced simplicial complexes giving a source of simplicial complexes exhibiting a Boolean decomposition similar to a geometric Lefschetz decomposition. The objects we are working with are $h$-vectors of flag spheres and balanced simplicial complexes whose $f$-vectors are equal to them. This builds on work of Nevo--Petersen--Tenner on a conjecture of Nevo--Petersen that the gamma vector of an odd-dimensional flag sphere is the $f$-vector of a balanced simplicial complex (which was shown for barycentric subdivisions by Nevo--Petersen--Tenner). We can connect our decomposition to positivity questions on reciprocal/palindromic polynomials associated to flag spheres and geometric questions motivating them. In addition, we note that the degrees in the Lefschetz-like decomposition are not halved unlike the usual $h$-vector setting. |
| title | $f$-vectors of balanced simplicial complexes, flag spheres, and geometric Lefschetz decompositions |
| topic | Combinatorics Algebraic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2410.08139 |