Browse issues
no. 2
$h$-vector inequalities under weak maps
Liu, Gaku1; Mason, Alexander
1 University of Washington Dept. of mathematics Seattle WA 98195-4350 (USA)
Algebraic Combinatorics, Volume 8 (2025) no. 2, pp. 479-493.

We study the behavior of $h$-vectors associated to matroid complexes under weak maps, or inclusions of matroid polytopes. Specifically, we show that the $h$-vector of the order complex of the lattice of flats of a matroid is component-wise non-increasing under a weak map. This result extends to the flag $h$-vector. We note that the analogous result also holds for independence complexes and rank-preserving weak maps.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.409
Classification: 05B35
Keywords: matroids, weak maps, $h$-vectors

Liu, Gaku 1; Mason, Alexander 

1 University of Washington Dept. of mathematics Seattle WA 98195-4350 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2025__8_2_479_0,
     author = {Liu, Gaku and Mason, Alexander},
     title = {$h$-vector inequalities under weak maps},
     journal = {Algebraic Combinatorics},
     pages = {479--493},
     publisher = {The Combinatorics Consortium},
     volume = {8},
     number = {2},
     year = {2025},
     doi = {10.5802/alco.409},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.409/}
}
TY  - JOUR
AU  - Liu, Gaku
AU  - Mason, Alexander
TI  - $h$-vector inequalities under weak maps
JO  - Algebraic Combinatorics
PY  - 2025
SP  - 479
EP  - 493
VL  - 8
IS  - 2
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.409/
DO  - 10.5802/alco.409
LA  - en
ID  - ALCO_2025__8_2_479_0
ER  - 
%0 Journal Article
%A Liu, Gaku
%A Mason, Alexander
%T $h$-vector inequalities under weak maps
%J Algebraic Combinatorics
%D 2025
%P 479-493
%V 8
%N 2
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.409/
%R 10.5802/alco.409
%G en
%F ALCO_2025__8_2_479_0
Liu, Gaku; Mason, Alexander. $h$-vector inequalities under weak maps. Algebraic Combinatorics, Volume 8 (2025) no. 2, pp. 479-493. doi : 10.5802/alco.409. https://alco.centre-mersenne.org/articles/10.5802/alco.409/

[1] Adiprasito, Karim; Huh, June; Katz, Eric Hodge theory for combinatorial geometries, Ann. of Math. (2), Volume 188 (2018) no. 2, pp. 381-452 | DOI | MR | Zbl

[2] Björner, Anders Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc., Volume 260 (1980) no. 1, pp. 159-183 | DOI | MR | Zbl

[3] Björner, Anders On the homology of geometric lattices, Algebra Universalis, Volume 14 (1982) no. 1, pp. 107-128 | DOI | MR

[4] Björner, Anders The homology and shellability of matroids and geometric lattices, Matroid applications (Encyclopedia Math. Appl.), Volume 40, Cambridge Univ. Press, Cambridge, 1992, pp. 226-283 | DOI | MR | Zbl

[5] Braden, Tom; Huh, June; Matherne, Jacob P.; Proudfoot, Nicholas; Wang, Botong Singular Hodge theory for combinatorial geometries, 2020 | arXiv | Zbl

[6] Elias, Ben; Miyata, Dane; Proudfoot, Nicholas; Vecchi, Lorenzo Categorical valuative invariants of polyhedra and matroids, 2024 | arXiv

[7] Ferroni, Luis; Matherne, Jacob P.; Stevens, Matthew; Vecchi, Lorenzo Hilbert-Poincaré series of matroid Chow rings and intersection cohomology, Adv. Math., Volume 449 (2024), Paper no. 109733, 55 pages | DOI | MR | Zbl

[8] Ferroni, Luis; Schröter, Benjamin Valuative invariants for large classes of matroids, J. Lond. Math. Soc. (2), Volume 110 (2024) no. 3, Paper no. e12984, 86 pages | DOI | MR | Zbl

[9] Lucas, Dean Weak maps of combinatorial geometries, Trans. Amer. Math. Soc., Volume 206 (1975), pp. 247-279 | DOI | MR | Zbl

[10] Nyman, Kathryn; Swartz, Ed Inequalities for the h-vectors and flag h-vectors of geometric lattices, Discrete Comput. Geom., Volume 32 (2004) no. 4, pp. 533-548 | DOI | MR | Zbl

[11] Oxley, James Matroid theory, Oxford Graduate Texts in Mathematics, 21, Oxford University Press, Oxford, 2011, xiv+684 pages | DOI | MR

[12] Stanley, Richard P. Balanced Cohen-Macaulay complexes, Trans. Amer. Math. Soc., Volume 249 (1979) no. 1, pp. 139-157 | DOI | MR | Zbl

[13] Stanley, Richard P. A monotonicity property of h-vectors and h * -vectors, European J. Combin., Volume 14 (1993) no. 3, pp. 251-258 | DOI | MR | Zbl

[14] Stanley, Richard P. Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR

[15] Sturmfels, Bernd On the matroid stratification of Grassmann varieties, specialization of coordinates, and a problem of N. White, Adv. Math., Volume 75 (1989) no. 2, pp. 202-211 | DOI | MR | Zbl

[16] Theory of matroids (White, Neil, ed.), Encyclopedia of Mathematics and its Applications, 26, Cambridge University Press, Cambridge, 1986, xviii+316 pages | DOI | MR

Cited by Sources: