Mathematics > Rings and Algebras
arXiv:2401.14648 (math)
[Submitted on 26 Jan 2024 (v1), last revised 25 Jul 2026 (this version, v3)]
Abstract:Given a graded bialgebra $H$, we let $\Delta^{\left[ k\right] }:H\rightarrow H^{\otimes k}$ and $m^{\left[ k\right] }:H^{\otimes k}\rightarrow H$ be its iterated (co)multiplications for all $k\in\mathbb{N}$. For any $k$-tuple $\alpha=\left(
\alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right) \in\mathbb{N}^{k}$ of nonnegative integers, and any permutation $\sigma$ of $\left\{ 1,2,\ldots,k\right\} $, we consider the map $p_{\alpha,\sigma}:=m^{\left[ k\right] }\circ P_{\alpha}\circ\sigma^{-1}\circ\Delta^{\left[ k\right] }:H\rightarrow H$, where $P_{\alpha}$ denotes the projection of $H^{\otimes k}$ onto its multigraded component $H_{\alpha_{1}}\otimes H_{\alpha_{2}}\otimes\cdots\otimes H_{\alpha_{k}}$, and where $\sigma^{-1}:H\rightarrow H$ permutes the tensor factors.
We prove formulas for the composition $p_{\alpha,\sigma}\circ p_{\beta,\tau}$ and the convolution $p_{\alpha,\sigma}\star p_{\beta,\tau}$ of two such maps. When $H$ is cocommutative, these generalize Patras's 1994 results (which, in turn, generalize Solomon's Mackey formula).
We also construct a combinatorial Hopf algebra $\operatorname*{PNSym}$ ("permuted noncommutative symmetric functions") that governs the maps $p_{\alpha,\sigma}$ for arbitrary connected graded bialgebras $H$ in the same way as the well-known $\operatorname*{NSym}$ governs them in the cocommutative case. We end by outlining an application to checking identities for connected graded Hopf algebras.
| Comments: | 85 pages. Partly an outline, partly an exposition. Submitted to the proceedings of CATMI 2023 Bergen. Comments are welcome! v3 fixes Theorem 1.37 and its proof (now claims slightly less) and replaces the wrong Theorem 2.10 by Remark 2.10. GPT-5.6 Sol found both mistakes, though I guess "proof idea" was a dead giveaway. Sorry! |
| Subjects: | Rings and Algebras (math.RA); Combinatorics (math.CO) |
| MSC classes: | 16T30, 05E05 |
| Cite as: | arXiv:2401.14648 [math.RA] |
| (or arXiv:2401.14648v3 [math.RA] for this version) | |
| https://doi.org/10.48550/arXiv.2401.14648 arXiv-issued DOI via DataCite |
Submission history
From: Darij Grinberg [view email]
[v1]
Fri, 26 Jan 2024 04:50:41 UTC (42 KB)
[v2]
Wed, 16 Oct 2024 19:49:09 UTC (70 KB)
[v3]
Sat, 25 Jul 2026 02:38:45 UTC (72 KB)
Current browse context:
math.RA
Bookmark
Bibliographic Tools
Bibliographic and Citation Tools
Bibliographic Explorer Toggle
Code, Data, Media
Code, Data and Media Associated with this Article
Demos
Demos
Related Papers
Recommenders and Search Tools
About arXivLabs
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.