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Peer-reviewedCC BY 4.0Seminaire Lotharingien de Combinatoire

Combinatorics of descent algebras and graph coverings

  • Philippe Biane
Open at sourcePDFCite

Abstract

We give a direct combinatorial proof that the product of two descent classes in a symmetric group is a sum of descent classes. The proof is based on the fact that the group product gives a covering map when descent classes are endowed with the graph structure coming from the weak order. The main geometric argument is valid for any Coxeter group, even infinite ones for which the descent algebra does not exist.

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Cite

BibTeXRIS
BibTeX
@article{biane2026combinatorics,
  title = {{Combinatorics of descent algebras and graph coverings}},
  author = {Philippe Biane},
  journal = {Seminaire Lotharingien de Combinatoire},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05349806v2},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Combinatorics of descent algebras and graph coverings
AU  - Philippe Biane
PY  - 2026
DA  - 2026/09/09
JO  - Seminaire Lotharingien de Combinatoire
UR  - https://hal.science/hal-05349806v2
AB  - We give a direct combinatorial proof that the product of two descent classes in a symmetric group is a sum of descent classes. The proof is based on the fact that the group product gives a covering map when descent classes are endowed with the graph structure coming from the weak order. The main geometric argument is valid for any Coxeter group, even infinite ones for which the descent algebra does not exist.
SN  - 1286-4889
ER  -