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Preprint
A Solomon Mackey formula for graded bialgebras
arXiv.org
26 Jan 2024
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.
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Details
- Title
- A Solomon Mackey formula for graded bialgebras
- Creators
- Darij Grinberg
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862234204721