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A Hopf algebra on permutations with a coupling product
Abstract
We define a coupling product and a draw coproduct on permutations and show that they define a graded, connected, cocommutative, free Hopf algebra ๐๐. In characteristic zero, this implies that ๐๐ is isomorphic to a certain cocommutative Hopf algebra associated with the dual of the coradical filtration of the Malvenuto--Reutenauer Hopf algebra on permutations. The coupling product and draw coproduct are permutation analogues of the product and coproduct on the monomial basis of symmetric functions; therefore, we can say that our presentation is a monomial basis for ๐๐.
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BibTeX
@misc{li2026hopf,
title = {{A Hopf algebra on permutations with a coupling product}},
author = {Huilan Li and Yishuo Yang and Nantel Bergeron},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.23345},
eprint = {2607.23345},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.23345},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A Hopf algebra on permutations with a coupling product
AU - Huilan Li
AU - Yishuo Yang
AU - Nantel Bergeron
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2607.23345
UR - https://arxiv.org/abs/2607.23345
AB - We define a coupling product and a draw coproduct on permutations and show that they define a graded, connected, cocommutative, free Hopf algebra ๐๐. In characteristic zero, this implies that ๐๐ is isomorphic to a certain cocommutative Hopf algebra associated with the dual of the coradical filtration of the Malvenuto--Reutenauer Hopf algebra on permutations. The coupling product and draw coproduct are permutation analogues of the product and coproduct on the monomial basis of symmetric functions; therefore, we can say that our presentation is a monomial basis for ๐๐.
ER -