The parity module of a matroid

@misc{braden2026parity,
  title = {{The parity module of a matroid}},
  author = {Tom Braden and June Huh and Jacob P. Matherne and Nicholas Proudfoot and Botong Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20234},
  eprint = {2610.20234},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20234},
}

TY  - UNPB
TI  - The parity module of a matroid
AU  - Tom Braden
AU  - June Huh
AU  - Jacob P. Matherne
AU  - Nicholas Proudfoot
AU  - Botong Wang
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20234
UR  - https://arxiv.org/abs/2610.20234
AB  - We introduce the parity module of a matroid with coefficients in a field k. When k has characteristic zero, we show that the parity module of a matroid is isomorphic to its intersection cohomology module, yielding a new intrinsic characterization of the latter. The structure and dimension of the parity module can vary depending on the characteristic, giving rise to characteristic-dependent generalizations of the Kazhdan-Lusztig and Z-polynomials of matroids. When the matroid is realizable over the complex numbers, the parity module is isomorphic to the hypercohomology of the parity sheaf supported on the arrangement Schubert variety of any such realization. We also introduce the notion of pure modules over any field k and show that they satisfy a matroid analogue of the Beilinson-Bernstein-Deligne-Gabber decomposition theorem: Every pure module over the graded Möbius algebra decomposes as a direct sum of shifts of the parity modules of matroids obtained by restricting to flats.
ER  -