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arxiv: 2401.11068 · v2 · submitted 2024-01-20 · 🧮 math.RT · math.GR

Defining sequences for fundamental root systems and Coxeter graphs for super Weyl groups

Changjie Chen , Yiyang Li , Bin Shu This is my paper

Pith reviewed 2026-05-24 04:47 UTC · model grok-4.3

classification 🧮 math.RT math.GR
keywords super Weyl groupsCoxeter graphsdefining sequencesfundamental root systemsLie superalgebrasclassical types A B C D
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The pith

Defining sequences for root systems determine the Coxeter graphs of super Weyl groups

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

The paper introduces defining sequences as a new description of fundamental root systems for the classical Lie superalgebras of types A, B, C, and D. These sequences are used to construct the Coxeter graphs that present the Coxeter groups of which the super Weyl groups are quotients. Super Weyl groups arise in the study of representations of basic classical Lie superalgebras and algebraic supergroups. The approach supplies an explicit combinatorial link between root data and the Coxeter presentation of these groups.

Core claim

Defining sequences provide a new description of fundamental root systems for basic classical Lie superalgebras of types A, B, C and D; from these sequences the Coxeter graphs of the associated super Weyl groups are read off, thereby identifying the Coxeter groups of which the super Weyl groups are quotients.

What carries the argument

Defining sequences, a new combinatorial description of fundamental root systems for types A, B, C, D, that generate the full set of relations for the Coxeter presentation of each super Weyl group.

Load-bearing premise

The super Weyl group is a quotient of a Coxeter group, and the newly defined sequences generate precisely the relations that recover that quotient.

What would settle it

An explicit computation, for a small-rank example such as type A(1,1) or B(1,1), showing that the group presented by the Coxeter graph read from the defining sequence is strictly larger or smaller than the super Weyl group defined in the cited reference.

read the original abstract

The super Weyl group of a basic classical Lie superalgebra was introduced and studied in \cite{PS}, which turns out to play an important role for the study of representations of the basic classical Lie superalgebras and algebraic supergroups (see \cite{PS, LS}). These groups turn out to be some quotients of Coxeter groups. It is deserved to specially investigate super Weyl groups via revealing the related Coxeter systems. The purpose of this paper is twofold. One is to describe the Coxeter systems for super Weyl groups of basic classical Lie superalgebras. The other one is to introduce defining sequences which are a kind of new descriptions of fundamental root systems for classical Lie superalgebras of type $A,B,C$ and $D$. Based on defining sequences, we decide the Coxeter groups associated with those super Weyl groups via Coxeter graphs.

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 introduces 'defining sequences' as a new description of fundamental root systems for classical Lie superalgebras of types A, B, C and D. It then uses these sequences to construct explicit Coxeter graphs that realize the super Weyl groups (previously shown in [PS] to be quotients of Coxeter groups) as Coxeter systems.

Significance. If the sequences are shown to generate precisely the required relations and the resulting graphs are verified to match the super Weyl groups, the constructions would supply concrete combinatorial presentations that could facilitate further study of representations of basic classical Lie superalgebras and algebraic supergroups.

minor comments (2)
  1. The abstract and introduction cite [PS] for the quotient property but do not indicate where in the manuscript the independence of the new defining sequences from that prior work is established.
  2. No explicit examples of defining sequences or the resulting Coxeter graphs appear in the provided abstract; the manuscript should include at least one fully worked example for type A or B to allow verification of the construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting its potential significance in providing concrete combinatorial presentations for super Weyl groups. The recommendation is listed as uncertain, but no specific major comments were provided in the report. We therefore have no point-by-point responses to address. We remain available to clarify any aspects of the constructions or provide additional verification if requested.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract describes the paper as introducing new defining sequences for fundamental root systems of classical Lie superalgebras and using them to determine associated Coxeter graphs for super Weyl groups, which were previously shown in the external citation [PS] to be quotients of Coxeter groups. No equations, self-citations, or constructions are provided that would allow any load-bearing step to reduce by definition or by construction to its own inputs. The work adds independent descriptive content rather than renaming or refitting prior results, satisfying the default expectation of no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the work appears to be a combinatorial re-description rather than an axiomatic extension.

pith-pipeline@v0.9.0 · 5679 in / 1092 out tokens · 19761 ms · 2026-05-24T04:47:39.325216+00:00 · methodology

discussion (0)

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Reference graph

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