We extend the definition of coarse flag Hilbert–Poincaré series to matroids; these series arise in the context of local Igusa zeta functions associated to hyperplane arrangements. We study these series in the case of oriented matroids by applying geometric and combinatorial tools related to their topes. In this case, we prove that the numerators of these series are coefficient-wise bounded below by the Eulerian polynomial and equality holds if and only if all topes are simplicial. Moreover this yields a sufficient criterion for non-orientability of matroids of arbitrary rank.
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Keywords: Coarse flag polynomial, Eulerian polynomials, Igusa zeta functions, oriented matroids
CC-BY 4.0
@article{ALCO_2023__6_3_623_0,
author = {K\"uhne, Lukas and Maglione, Joshua},
title = {On the geometry of flag {Hilbert{\textendash}Poincar\'e} series for matroids},
journal = {Algebraic Combinatorics},
pages = {623--638},
publisher = {The Combinatorics Consortium},
volume = {6},
number = {3},
year = {2023},
doi = {10.5802/alco.276},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.276/}
}
TY - JOUR AU - Kühne, Lukas AU - Maglione, Joshua TI - On the geometry of flag Hilbert–Poincaré series for matroids JO - Algebraic Combinatorics PY - 2023 SP - 623 EP - 638 VL - 6 IS - 3 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.276/ DO - 10.5802/alco.276 LA - en ID - ALCO_2023__6_3_623_0 ER -
%0 Journal Article %A Kühne, Lukas %A Maglione, Joshua %T On the geometry of flag Hilbert–Poincaré series for matroids %J Algebraic Combinatorics %D 2023 %P 623-638 %V 6 %N 3 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.276/ %R 10.5802/alco.276 %G en %F ALCO_2023__6_3_623_0
Kühne, Lukas; Maglione, Joshua. On the geometry of flag Hilbert–Poincaré series for matroids. Algebraic Combinatorics, Volume 6 (2023) no. 3, pp. 623-638. doi : 10.5802/alco.276. https://alco.centre-mersenne.org/articles/10.5802/alco.276/
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