A Solomon Mackey formula for graded bialgebras

Abstract

Given a graded bialgebra $H$, we let ${\Delta }^{\left[k\right]}:H\to H^{\otimes k}$ and $m^{\left[k\right]}:H^{\otimes k}\to H$ be its iterated (co)multiplications for all $k\in \mathbb{N}$. For any $k$-tuple $\alpha =\left({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k\right)\in {\mathbb{N}}^k$ of nonnegative integers, and any permutation $\sigma$ of $\left\{1,2,\dots ,k\right\}$, we consider the map $p_{\alpha ,\sigma }:=m^{\left[k\right]}\circ P_{\alpha }\circ {\sigma }^{-1}\circ {\Delta }^{\left[k\right]}:H\to H$, 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 ${\sigma }^{-1}:H\to H$ 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 $\mathrm{PNSym}$ ("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 $\mathrm{NSym}$ governs them in the cocommutative case. We end by outlining an application to checking identities for connected graded Hopf algebras.

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!