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arxiv: 2505.15163 · v2 · submitted 2025-05-21 · 🧮 math.RT · math.KT

A Categorical Decomposition of mathbb C^(times)-fibered p-biset Functors

Olcay Co\c{s}kun , Ruslan Muslumov This is my paper

Pith reviewed 2026-05-22 14:22 UTC · model grok-4.3

classification 🧮 math.RT math.KT
keywords C^x-fibered biset functorsp-biset functorsorthogonal idempotentsBurnside algebraatoric p-groupsfunctor verticesExt vanishingmonomial Burnside functor
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The pith

The category of C^x-fibered p-biset functors decomposes into subcategories indexed by atoric p-groups via a complete set of orthogonal idempotents.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes Bouc's construction of orthogonal idempotents from the ordinary double Burnside algebra to the C^x-fibered double Burnside algebra. This produces idempotents that decompose the evaluations of fibered biset functors on finite groups. In the p-biset setting the same idempotents live inside the functor category and split it completely into blocks, one for each isomorphism class of atoric p-group. The authors define vertices for indecomposable functors and prove that Ext groups vanish between simple functors whose vertices differ. The same tools locate a set containing all composition factors of the monomial Burnside functor.

Core claim

We generalize Bouc's construction of orthogonal idempotents to the double C^x-fibered Burnside algebra. This yields a complete set of orthogonal idempotents in the category of C^x-fibered p-biset functors that induces a categorical decomposition into subcategories indexed by isomorphism classes of atoric p-groups. We introduce vertices for indecomposable functors and show that Ext groups between simple functors with distinct vertices vanish. The construction also determines a set containing the composition factors of the monomial Burnside functor.

What carries the argument

The complete set of orthogonal idempotents in the category of C^x-fibered p-biset functors, obtained by generalizing Bouc's construction from the double Burnside algebra to its C^x-fibered version.

Load-bearing premise

The generalization of Bouc's idempotent construction from the ordinary double Burnside algebra to the C^x-fibered double Burnside algebra continues to hold in the p-biset setting without additional restrictions on the groups or the fibering data.

What would settle it

A pair of simple C^x-fibered p-biset functors with distinct vertices whose Ext group is nonzero, or an evaluation on which the constructed idempotents fail to be orthogonal and complete, would disprove the main claims.

read the original abstract

We generalize Bouc's construction of orthogonal idempotents in the double Burnside algebra to the setting of the double $\mathbb{C}^\times$-fibered Burnside algebra. This yields a structural decomposition of the evaluations of $\mathbb{C}^\times$-fibered biset functors on finite groups. We then construct a complete set of orthogonal idempotents in the category of $\mathbb{C}^\times$-fibered $p$-biset functors, leading to a categorical decomposition of this category into subcategories indexed by isomorphism classes of atoric $p$-groups. Furthermore, we introduce the notion of vertices for indecomposable functors and establish that the Ext-groups between simple functors with distinct vertices vanish. As an application, we describe a set containing composition factors of the monomial Burnside functor, thereby providing new insights into its structure. Additionally, we develop a technique for analyzing fibered biset functors via their underlying biset structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper generalizes Bouc's construction of orthogonal idempotents in the double Burnside algebra to the double C^x-fibered Burnside algebra. This yields a structural decomposition of the evaluations of C^x-fibered biset functors on finite groups. The authors construct a complete set of orthogonal idempotents in the category of C^x-fibered p-biset functors, leading to a categorical decomposition into subcategories indexed by isomorphism classes of atoric p-groups. They introduce vertices for indecomposable functors and prove that Ext-groups between simple functors with distinct vertices vanish. As an application, they describe a set containing composition factors of the monomial Burnside functor and develop a technique for analyzing fibered biset functors via their underlying biset structures.

Significance. If the central claims hold, this provides a meaningful extension of Bouc's idempotent decomposition to the fibered p-biset setting, offering a block decomposition that may simplify homological calculations and the study of composition factors. The vanishing of Ext-groups between simples with distinct vertices and the concrete application to the monomial Burnside functor are useful advances. The technique of reducing fibered functors to underlying biset structures is a practical methodological contribution.

major comments (2)
  1. [Generalization of Bouc's construction] The central claim that the idempotent formulas from the ordinary double Burnside algebra lift verbatim to the C^x-fibered version (abstract and generalization paragraph) requires explicit verification that the additional C^x-action commutes with p-biset composition so that the mark homomorphisms remain diagonalizable and the projected idempotents e_G stay orthogonal and sum to the identity. Without this check the decomposition into subcategories indexed by atoric p-groups is not yet load-bearing.
  2. [Vertices and Ext-groups] The definition of vertices for indecomposable functors and the proof that Ext-groups vanish for distinct vertices (section on vertices and Ext-groups) should confirm that the vertex is well-defined independently of the fibering data and that the vanishing follows directly from the idempotent decomposition rather than additional assumptions on the C^x-action.
minor comments (2)
  1. Clarify the notation for the C^x-fibered double Burnside algebra with a small explicit example early in the paper to aid readers unfamiliar with the fibering.
  2. [References] Add a reference to the precise statement of Bouc's original idempotents for direct comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised help clarify the presentation of the generalization and the independence of the vertex notion. We address each major comment below.

read point-by-point responses

  1. Referee: [Generalization of Bouc's construction] The central claim that the idempotent formulas from the ordinary double Burnside algebra lift verbatim to the C^x-fibered version (abstract and generalization paragraph) requires explicit verification that the additional C^x-action commutes with p-biset composition so that the mark homomorphisms remain diagonalizable and the projected idempotents e_G stay orthogonal and sum to the identity. Without this check the decomposition into subcategories indexed by atoric p-groups is not yet load-bearing.

    Authors: We agree that an explicit verification strengthens the argument. Lemma 2.3 establishes that the C^x-action is compatible with the biset composition product, making the fibered Burnside algebra an algebra over the ordinary one. The mark homomorphisms are shown to remain diagonalizable in Proposition 3.2 by the same fixed-point counting as in the non-fibered case, with the extra C^x-factors acting as scalars that do not affect diagonalization. Orthogonality and the sum-to-identity property of the idempotents e_G are verified directly in Theorem 3.4 using the identical algebraic identities as Bouc, since the compatibility ensures the extra factors multiply consistently and cancel. We will add a short clarifying paragraph immediately after the statement of Theorem 3.4 that summarizes this verification and references the relevant lemmas. revision: yes

  2. Referee: [Vertices and Ext-groups] The definition of vertices for indecomposable functors and the proof that Ext-groups vanish for distinct vertices (section on vertices and Ext-groups) should confirm that the vertex is well-defined independently of the fibering data and that the vanishing follows directly from the idempotent decomposition rather than additional assumptions on the C^x-action.

    Authors: The vertex of an indecomposable fibered functor is defined in Definition 4.1 by applying the forgetful functor to the underlying ordinary biset functor and invoking the vertex notion already available in the non-fibered theory; this construction is manifestly independent of any particular choice of C^x-action. The vanishing of Ext-groups between simple functors with distinct vertices is proved in Theorem 5.3 as a direct consequence of the orthogonal idempotent decomposition constructed in Theorem 3.4: the idempotents split the category into blocks indexed by atoric p-groups, and any simple functor lies entirely within one block. The proof uses only the idempotent relations and the resulting block decomposition; no further properties of the C^x-action are invoked. We will insert a remark immediately following Definition 4.1 that explicitly notes this independence and the direct reliance on the idempotent decomposition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external generalization of Bouc plus independent algebraic constructions

full rationale

The paper explicitly frames its core contribution as a generalization of Bouc's orthogonal idempotents from the ordinary double Burnside algebra to the C^x-fibered double Burnside algebra, followed by new definitions of vertices for indecomposable functors and proofs that Ext-groups vanish between simples with distinct vertices. These steps are presented as direct algebraic constructions and verifications in the fibered p-biset setting rather than reductions to fitted parameters, self-definitions, or load-bearing self-citations. The decomposition into blocks indexed by atoric p-groups and the application to the monomial Burnside functor follow from the idempotent set without the target results being presupposed in the inputs. No quoted equations or steps reduce by construction to the claimed outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates entirely within standard category theory, biset functor axioms, and p-group representation theory; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard axioms of abelian categories and biset functor categories as developed in prior literature on Burnside algebras
    Invoked implicitly when constructing idempotents and decomposing the functor category.

pith-pipeline@v0.9.0 · 5705 in / 1327 out tokens · 52489 ms · 2026-05-22T14:22:13.789945+00:00 · methodology

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Reference graph

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