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arxiv: 2605.27738 · v1 · pith:7KRMLV25new · submitted 2026-05-26 · 🧮 math.RT · math.RA

Two-term tilting complexes of biserial fractional Brauer graph algebras

Bohan Xing This is my paper

Pith reviewed 2026-06-29 14:15 UTC · model grok-4.3

classification 🧮 math.RT math.RA
keywords biserial fractional Brauer graph algebrastilting-discretetwo-term tilting complexesBrauer graph algebrasderived equivalenceKauer movesskew group algebras
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The pith

Biserial fractional Brauer graph algebras are tilting-discrete exactly when their reduced Brauer graph forms are.

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

The paper establishes that a biserial fractional Brauer graph algebra has the tilting-discrete property if and only if the ordinary Brauer graph algebra obtained by reducing it does. It shows this by relating two-term tilting complexes and Kauer moves across the reduction, and additionally proves that skew group algebras induced by the Nakayama automorphism are skew-Brauer graph algebras. The work further shows that the tilting-discrete members of this class remain tilting-discrete after any derived equivalence. A reader would care because the result reduces questions about tilting discreteness in the generalized fractional setting to the classical Brauer graph case.

Core claim

A biserial fractional Brauer graph algebra is tilting-discrete if and only if its reduced form, which is a Brauer graph algebra, is tilting-discrete. Tilting-discrete biserial fractional Brauer graph algebras are closed under derived equivalence.

What carries the argument

The reduction operation that produces an ordinary Brauer graph algebra from a biserial fractional one, and the resulting equivalence on tilting-discrete status.

If this is right

  • Tilting-discrete status transfers exactly across the reduction in either direction.
  • Two-term tilting complexes of the fractional algebra correspond to those of the reduced algebra.
  • Any classification of tilting-discrete Brauer graph algebras immediately classifies the fractional versions.
  • The tilting-discrete property is invariant under derived equivalence for these algebras.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Known results on tilting discreteness for classical Brauer graph algebras apply verbatim to the fractional case.
  • Kauer moves on the reduced algebra may lift to produce corresponding moves on the fractional algebra.
  • Tilting discreteness behaves as a derived invariant within this family of algebras.

Load-bearing premise

The reduction from a biserial fractional Brauer graph algebra to its ordinary Brauer graph form preserves the tilting-discrete property in both directions.

What would settle it

An explicit biserial fractional Brauer graph algebra whose reduced Brauer graph form has a different tilting-discrete status than the original algebra.

read the original abstract

Brauer graph algebras form a classical class of symmetric algebras with well-structured combinatorial properties and geometric models. Recently, they have been generalized to biserial fractional Brauer graph algebras, which can be regarded as a self-injective version of the classical Brauer graph algebras. In this paper, we show that the skew group algebras of biserial fractional Brauer graph algebras induced by the Nakayama automorphism are in fact skew-Brauer graph algebras. We then study two-term tilting complexes and Kauer moves for biserial fractional Brauer graph algebras. Moreover, we prove that a biserial fractional Brauer graph algebra is tilting-discrete if and only if its reduced form (which is a Brauer graph algebra) is tilting-discrete. Finally, we show that tilting-discrete biserial fractional Brauer graph algebras are closed under derived equivalence.

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

0 major / 2 minor

Summary. The manuscript shows that skew group algebras of biserial fractional Brauer graph algebras induced by the Nakayama automorphism are skew-Brauer graph algebras. It examines two-term tilting complexes and Kauer moves on biserial fractional Brauer graph algebras. The central result is the if-and-only-if statement that such an algebra is tilting-discrete precisely when its reduced form (an ordinary Brauer graph algebra) is tilting-discrete. It concludes by proving that the tilting-discrete biserial fractional Brauer graph algebras are closed under derived equivalence.

Significance. If the reduction correspondence is correctly established, the work reduces the tilting-discreteness question for the larger class to the already-understood case of Brauer graph algebras and establishes invariance of the property under derived equivalence. This supplies a concrete combinatorial bridge between the two families and enlarges the set of algebras for which tilting-discreteness can be decided by reduction to the classical setting.

minor comments (2)
  1. [Abstract] The abstract asserts the equivalence via reduction without naming the section or theorem that contains the explicit correspondence between two-term tilting complexes before and after reduction; adding a forward reference would improve readability.
  2. Notation for the reduction map and the induced map on tilting complexes should be introduced once in a dedicated subsection and then used consistently; scattered definitions make the argument harder to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of the manuscript, as well as the recommendation of minor revision. No major comments appear in the provided report, so we have no specific points requiring point-by-point response at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes an if-and-only-if equivalence between tilting-discreteness of biserial fractional Brauer graph algebras and their reduced Brauer graph algebra forms, plus closure under derived equivalence. This rests on an explicit reduction map and prior combinatorial understanding of ordinary Brauer graph algebras, without any quoted step that defines the target property in terms of itself or renames a fitted input as a prediction. No load-bearing self-citation chains or ansatz smuggling appear in the described argument structure. The derivation is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper rests on the standard definitions and combinatorial models of Brauer graph algebras and tilting complexes from the representation-theory literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

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