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arxiv: 2606.11551 · v2 · pith:J3BSED7Unew · submitted 2026-06-10 · 🧮 math.RT

Gelfand--Kirillov dimensions of highest weight modules for basic classical Lie superalgebras

Jing Jiang This is my paper

Pith reviewed 2026-06-27 08:12 UTC · model grok-4.3

classification 🧮 math.RT
keywords Gelfand-Kirillov dimensionhighest weight modulesLie superalgebrascombinatorial algorithmLusztig a-functionRobinson-Schensted algorithmsl(m|n)osp(2|2n)
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The pith

The Gelfand-Kirillov dimension of simple highest weight modules over basic classical Lie superalgebras is fixed entirely by the even subalgebra.

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

This paper develops a combinatorial algorithm to compute the Gelfand-Kirillov dimension of simple highest weight modules for basic classical Lie superalgebras. It extends Lusztig's a-function and the Robinson-Schensted insertion algorithm from the classical Lie algebra setting to the superalgebra families sl(m|n) and osp(2|2n). The central result states that this dimension depends only on the even part of the superalgebra. Readers would care because the result reduces a superalgebra computation to an ordinary Lie algebra one.

Core claim

We develop a combinatorial algorithm to compute the Gelfand--Kirillov dimension of simple highest weight modules for basic classical Lie superalgebras. Building upon the results for classical Lie algebras via Lusztig's a-function and the Robinson--Schensted insertion algorithm, we extend these techniques to the super setting, providing explicit formulas for types sl(m|n) and osp(2|2n). Our results show that the GK dimension of a simple highest weight module is determined entirely by the even part of the Lie superalgebras.

What carries the argument

The combinatorial algorithm that applies Lusztig's a-function and the Robinson-Schensted insertion algorithm to the super setting for types sl(m|n) and osp(2|2n), which produces the GK dimension from data of the even subalgebra alone.

If this is right

  • Explicit formulas for the GK dimensions are obtained for simple highest weight modules of types sl(m|n) and osp(2|2n).
  • The GK dimension of any such module equals the GK dimension of the corresponding module over the even subalgebra.
  • The algorithm computes the dimension using only even-root data and the classical combinatorial tools.

Where Pith is reading between the lines

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

  • The same extension might be checked for the remaining basic classical types if the insertion rules can be defined without odd-root corrections.
  • The result indicates that the asymptotic growth rate of these modules is insensitive to the odd structure of the superalgebra.
  • It raises the question of whether other numerical invariants of highest weight modules over these superalgebras likewise reduce to the even subalgebra.

Load-bearing premise

The combinatorial techniques from classical Lie algebras via Lusztig's a-function and the Robinson-Schensted insertion algorithm extend directly to the super setting for types sl(m|n) and osp(2|2n) without modification for odd roots.

What would settle it

An independent calculation of the GK dimension for a simple highest weight module over sl(2|1) that differs from the value obtained by applying the even-subalgebra algorithm to its even part would falsify the claim.

read the original abstract

In this paper we develop a combinatorial algorithm to compute the Gelfand--Kirillov (GK) dimension of simple highest weight modules for basic classical Lie superalgebras. Building upon the results for classical Lie algebras via Lusztig's {\bf a}-function and the Robinson--Schensted (RS) insertion algorithm, we extend these techniques to the super setting, providing explicit formulas for types $\mathfrak{sl}(m|n)$ and $\mathfrak{osp}(2|2n)$. Our results show that the GK dimension of a simple highest weight module is determined entirely by the even part of the Lie superalgebras.

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 / 1 minor

Summary. This manuscript develops a combinatorial algorithm to compute the Gelfand-Kirillov dimensions of simple highest weight modules for basic classical Lie superalgebras. Building on Lusztig's a-function and the Robinson-Schensted insertion algorithm from the classical Lie algebra setting, it extends these methods to the superalgebra context and supplies explicit formulas for the types sl(m|n) and osp(2|2n). The central conclusion is that the GK dimension of such a module is determined entirely by the even part of the Lie superalgebra.

Significance. If the claimed extension holds, the paper would supply a concrete computational procedure for these dimensions in the super setting and establish that the odd root system contributes nothing, which could simplify many calculations in the representation theory of basic classical Lie superalgebras.

major comments (2)
  1. [Abstract] Abstract: the claim that the GK dimension depends only on the even part rests on the assertion that the a-function and RS insertion computed from the even roots alone suffice without adjustment; the manuscript provides no argument showing that the odd positive roots leave the relevant weight poset, Weyl group orbits, and insertion tableaux unchanged relative to the even subalgebra.
  2. [The extension to superalgebras] The extension section (implicit in the abstract's description of the algorithm): no low-rank explicit comparison is given between the superalgebra case and its even subalgebra to verify that the odd roots contribute zero to the a-function value or RS output; such a check is load-bearing for the independence claim.
minor comments (1)
  1. The abstract states that explicit formulas are provided for sl(m|n) and osp(2|2n) but does not indicate whether the method is intended to apply to the remaining basic classical types; this scope should be clarified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. The major comments identify places where additional explicit justification and verification would strengthen the presentation of the independence claim. We respond point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the GK dimension depends only on the even part rests on the assertion that the a-function and RS insertion computed from the even roots alone suffice without adjustment; the manuscript provides no argument showing that the odd positive roots leave the relevant weight poset, Weyl group orbits, and insertion tableaux unchanged relative to the even subalgebra.

    Authors: We agree that the manuscript would benefit from an explicit argument establishing that the odd positive roots leave the weight poset, Weyl group orbits, and RS insertion tableaux unchanged. While the combinatorial construction in the body of the paper proceeds entirely from the even roots, a dedicated clarifying paragraph will be added immediately after the statement of the main algorithm to prove this invariance, using the fact that the odd roots do not intersect the even weight lattice or alter the relevant Bruhat order. revision: yes

  2. Referee: [The extension to superalgebras] The extension section (implicit in the abstract's description of the algorithm): no low-rank explicit comparison is given between the superalgebra case and its even subalgebra to verify that the odd roots contribute zero to the a-function value or RS output; such a check is load-bearing for the independence claim.

    Authors: We accept that low-rank explicit comparisons are a useful verification. The revised manuscript will contain a new short subsection (or appendix) providing direct computations for the cases sl(2|1) and osp(2|2), confirming that both the a-function values and the RS insertion tableaux coincide exactly with those obtained from the even subalgebra alone. revision: yes

Circularity Check

0 steps flagged

No circularity: extension of classical a-function/RS methods is independent of target result

full rationale

The paper states it builds upon established results for classical Lie algebras using Lusztig's a-function and the Robinson-Schensted algorithm, then extends these to the super setting for sl(m|n) and osp(2|2n) to obtain explicit formulas. The claim that GK dimension is determined by the even part follows from this extension rather than from any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. No equations or steps reduce the output to the input by construction, and the derivation remains self-contained against the cited classical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, axioms, or invented entities can be identified from the provided information.

pith-pipeline@v0.9.1-grok · 5627 in / 1173 out tokens · 30590 ms · 2026-06-27T08:12:15.662748+00:00 · methodology

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