🔢 mathematics

A Solomon Mackey formula for graded bialgebras

This paper establishes generalized Solomon Mackey formulas for the composition and convolution of specific maps on graded bialgebras, extending known cocommutative results to the general case through the construction of a new combinatorial Hopf algebra called PNSym.

Original authors: Darij Grinberg

Published 2026-07-28

📖 4 min read🧠 Deep dive

CC0 1.0

Original authors: Darij Grinberg

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a universe built not of stars and planets, but of invisible building blocks called "algebras." In this world, mathematicians study how these blocks can be stacked, split, and rearranged. Two of the most popular ways to play with these blocks are "multiplication" (gluing them together) and "comultiplication" (splitting them apart). When you have a system where you can do both at the same time, you get something called a bialgebra. If this system also has a special "undo" button that lets you reverse the splitting, it becomes a Hopf algebra. These aren't just abstract toys; they are the hidden grammar behind everything from quantum physics to the way we count complex patterns in nature.

For decades, mathematicians have been trying to write down the "rules of the road" for these algebras. Specifically, they wanted to know: if you take a block, split it, shuffle the pieces around, glue them back together, and then split them again, what happens? In the 1990s, a brilliant formula was discovered for a very specific, tidy version of these algebras (where the order of splitting doesn't matter). It was like finding a perfect recipe for a cake that only works if you use a specific brand of flour. But what if you use a different flour? What if the order of your ingredients actually changes the taste? For a long time, no one knew the recipe for the messy, general case.

This paper, written by Darij Grinberg, is like a master chef finally writing down the recipe for every kind of flour, not just the tidy kind. The author introduces a new, slightly chaotic set of operations called "twisted projecting Adams operations." Think of these as a game where you take a pile of blocks, split them into a specific number of piles, shuffle those piles in a specific order, filter out the ones that are the wrong size, and then glue them back together. The paper proves a massive, general formula that tells you exactly what happens if you play this game twice in a row. It turns out the answer is a complex sum of many different ways the blocks could have been rearranged, governed by a new mathematical structure the author calls PNSym (Permuted Noncommutative Symmetric Functions).

The author doesn't just stop at the recipe; they build a whole new kitchen to store it. They construct a "combinatorial Hopf algebra" named PNSym, which acts as a universal control center for these operations. Just as a map helps you navigate a city, PNSym helps mathematicians navigate the complex relationships between these algebraic operations. The paper shows that this new structure is "self-opposite," meaning it has a built-in symmetry that allows you to reverse the process perfectly, a property that the older, simpler version of these algebras lacked.

The findings are rigorous and proven, not just guessed. The author demonstrates that these operations are "linearly independent," which is a fancy way of saying that every single move in this game produces a unique result that cannot be faked by mixing other moves together. This means the new formula is the only way to describe these operations correctly. The paper also outlines a practical application: a mechanical algorithm that can check if any proposed identity (a rule about how these blocks behave) is true or false for any connected graded Hopf algebra. While the author admits the proofs are a bit rough around the edges and the field is still evolving, the core discovery—that a general formula exists and that PNSym is the key to unlocking it—is presented as a solid, new foundation for the field. It's a bit like realizing that the chaotic jumble of a messy room actually follows a hidden, beautiful pattern, and finally having the blueprint to organize it.

Technical Summary: A Solomon Mackey Formula for Graded Bialgebras

Problem Statement
The paper addresses the classification of natural $k$ -linear operations on the category of graded $k$ -bialgebras. Specifically, it investigates the algebraic structure formed by "twisted projecting Adams operations," denoted $p_{\alpha, \sigma}$ . These operators are defined for a graded bialgebra $H$ using iterated comultiplications $\Delta^{[k]}$ , iterated multiplications $m^{[k]}$ , projections $P_\alpha$ onto multigraded components, and permutations $\sigma \in S_k$ acting on tensor factors.

While the composition of such operators is well-understood in the special cases where $H$ is commutative or cocommutative (recovering Solomon's Mackey formula for the symmetric group and Patras's results on descent operators), the general case for arbitrary graded bialgebras remained open. In the general case, the set of operators $p_{\alpha, \sigma}$ is closed under composition, but the formula for $p_{\alpha, \sigma} \circ p_{\beta, \tau}$ was previously unknown. Furthermore, the paper seeks to identify the combinatorial Hopf algebra that governs these operations for arbitrary connected graded bialgebras, analogous to how the algebra of noncommutative symmetric functions ( $\text{NSym}$ ) governs them in the cocommutative case.

Methodology
The author employs a combination of direct algebraic manipulation and combinatorial construction:

Algebraic Derivation: The core of the work involves deriving an explicit formula for the composition $p_{\alpha, \sigma} \circ p_{\beta, \tau}$ . This requires establishing a series of commutation relations between the constituent maps: projections ( $P_\alpha$ ), permutations ( $\sigma$ ), iterated multiplications ( $m^{[k]}$ ), and iterated comultiplications ( $\Delta^{[k]}$ ). Key technical tools include the "Zolotarev shuffle" permutation and specific lemmas regarding how these operators interact with tensor products and graded components.
Linear Independence: To validate the generality of the derived formulas, the author constructs specific "generic" connected graded Hopf algebras (free algebras with specific comultiplication rules) to demonstrate that the family of operators $\{p_{\alpha, \sigma}\}$ is $k$ -linearly independent. This ensures that the derived composition formulas are not artifacts of specific algebraic relations in a particular $H$ but hold universally.
Combinatorial Construction: The paper constructs a new combinatorial Hopf algebra, named PNSym (Permuted Noncommutative Symmetric Functions). The basis of this algebra is indexed by "mopiscotions" (permuted compositions), which are pairs $(\alpha, \sigma)$ consisting of a composition $\alpha$ and a permutation $\sigma$ . The algebraic structure of PNSym is defined to mirror the composition, convolution, and tensor product properties of the operators $p_{\alpha, \sigma}$ .
Connection to Monoid Algebras: The internal multiplication of PNSym is analyzed by relating it to the Janus monoid (introduced by Aguiar and Mahajan) and the face monoid of the braid arrangement. The author establishes that the algebra of operators on the $n$ -th graded component is isomorphic to the $S_n$ -invariant part of the Janus algebra.

Key Contributions and Results

The Generalized Solomon Mackey Formula (Theorem 1.19): The primary result is an explicit formula for the composition of two twisted projecting Adams operations:
$p_{\alpha, \sigma} \circ p_{\beta, \tau} = \sum_{\gamma} p_{\gamma, \tau[\sigma]}$
where the sum runs over all $k \times \ell$ matrices of nonnegative integers $\gamma_{i,j}$ satisfying specific row and column sum constraints determined by $\alpha$ and $\beta$ . The resulting permutation is given by a specific operation $\tau[\sigma]$ (a wreath-product-like composition of permutations). This formula generalizes Patras's 1994 results and Solomon's Mackey formula to arbitrary graded bialgebras.
Convolution and Tensor Formulas: The paper provides formulas for the convolution product ( $p_{\alpha, \sigma} \star p_{\beta, \tau} = p_{\alpha\beta, \sigma \oplus \tau}$ ) and the behavior of these operators under tensor products of bialgebras and duality.
The Algebra PNSym: A new combinatorial Hopf algebra, PNSym, is constructed.
- It possesses an "external" multiplication (mirroring convolution) and an "internal" multiplication (mirroring composition).
- It is shown to be a connected graded cocommutative Hopf algebra under the external product and a non-unital bialgebra under the internal product.
- There exists a surjective morphism $p: \text{PNSym} \to \text{NSym}$ that respects all structures, effectively showing that PNSym is a "permuted" lift of the classical noncommutative symmetric functions.
Linear Independence: Theorem 1.36 establishes that for a suitably chosen connected graded Hopf algebra, the operators $p_{\alpha, \sigma}$ are linearly independent. This confirms that the derived formulas are the most general possible identities.
Isomorphism to Invariant Subalgebras: The paper proves that the algebra of operators on the $n$ -th graded component, $\text{PNSym}^{(2)}_n$ , is isomorphic to the $S_n$ -invariant subalgebra of the Janus algebra (Theorem 2.18 and Corollary 2.19). This connects the study of these operators to the rich theory of the descent algebra and the Tits monoid.

Significance and Claims
The author presents this work as a preliminary report on a project aiming to classify identities between natural operations on graded bialgebras. The paper claims that:

The derived composition formula (Theorem 1.19) is a new result that generalizes known formulas for cocommutative and commutative cases to the general setting.
The combinatorial Hopf algebra PNSym is a new object that "governs" the twisted projecting Adams operations for arbitrary connected graded bialgebras, just as $\text{NSym}$ governs them in the cocommutative case.
The results provide a mechanical algorithm for verifying functorial identities for connected graded Hopf algebras by reducing them to calculations within the combinatorial algebra PNSym.

The author explicitly notes that the proofs are "inchoate" and lack elegance, and that the focus is currently restricted to unary operations ( $H \to H$ ), leaving multi-input-multi-output operations for future study. The paper also leaves open the question of whether all natural graded endomorphisms of connected graded bialgebras can be expressed as infinite linear combinations of the $p_{\alpha, \sigma}$ operators (Question 1.41), though the author suspects the answer is affirmative.

More like this