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arxiv: 2412.15346 · v3 · submitted 2024-12-19 · 🧮 math.RT

Howe duality over finite fields I: The two stable ranges

Sophie Kriz This is my paper

Pith reviewed 2026-05-23 07:07 UTC · model grok-4.3

classification 🧮 math.RT
keywords Howe dualityfinite fieldsoscillator representationstable rangessymplectic grouporthogonal grouprepresentation restriction
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The pith

The Howe duality correspondence is constructed explicitly in the stable ranges over finite fields.

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

The paper establishes an explicit construction of the Howe duality correspondence arising from the restriction of an oscillator representation of the symplectic group to a product of symplectic and orthogonal groups over finite fields. This is achieved specifically in the two stable ranges where the rank of one group is large enough compared to the other. A sympathetic reader would care because the construction gives a concrete description of how representations correspond in these cases, setting up the series for a full explicit description and for proving results on multiplicities and characters.

Core claim

This paper constructs the correspondence in the two so called stable ranges, where the rank of one of the factors is large enough with respect to the other, by providing an explicit description of the restriction of the oscillator representation to the product subgroup under the stable range condition.

What carries the argument

The oscillator representation of the symplectic group and its restriction to the product of symplectic and orthogonal subgroups under the stable range condition.

If this is right

  • The correspondence pairs representations of the symplectic and orthogonal groups explicitly in the stable ranges.
  • The restriction of the oscillator representation decomposes according to this pairing when one rank is large enough relative to the other.
  • The construction provides the base case for extending the description to all ranks.

Where Pith is reading between the lines

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

  • This construction allows later demonstration that tensor pairs occur with multiplicity one.
  • It supports proving the type C case of the Gurevich-Howe rank conjecture.
  • A recursive formula for the characters of cuspidal unipotent representations can be obtained from the full series.

Load-bearing premise

The oscillator representation of the symplectic group exists and its restriction to the product subgroup admits an explicit description precisely when one rank dominates the other.

What would settle it

A direct computation of the restriction decomposition for a small finite field and concrete ranks satisfying the stable range condition that fails to match the constructed pairing or multiplicities.

read the original abstract

This is the first in a series of papers on type I Howe duality for finite fields, concerning the restriction of an oscillator representation of the symplectic group to a product of a symplectic and an orthogonal group. The goal of the series is describing this restriction completely explicitly. Applications (described in the third paper of the series) include demonstrating that the tensor pairs previously calculated by S.-Y. Pan as occuring with non-zero multiplicity occur with multiplicity 1, proving the type C case of the Gurevich-Howe rank conjecture, and giving a recursive formula for the characters of cuspidal unipotent representations. In this first paper, we construct the correspondence in the two so called stable ranges, where the rank of one of the factors is large enough with respect to the other.

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 / 3 minor

Summary. The paper constructs an explicit description of the Howe correspondence arising from the restriction of the oscillator representation of the symplectic group over a finite field to a product of a symplectic group and an orthogonal group, precisely in the two stable ranges (where the rank of one factor is large enough relative to the other). This is the first paper in a series whose later installments will address the non-stable cases and derive applications including multiplicity-one statements for certain tensor pairs, the type-C case of the Gurevich-Howe rank conjecture, and recursive character formulas for cuspidal unipotent representations.

Significance. If the claimed explicit construction holds, the result supplies the base case needed for the remainder of the series and thereby enables the listed applications. The work is situated in the representation theory of finite groups of Lie type and supplies a concrete, range-dependent description of a classical theta correspondence that had previously been studied primarily over local fields or in the complex case.

minor comments (3)
  1. The abstract states that the construction is given 'precisely when one rank dominates the other,' but the introduction should include a precise numerical statement of the two stable-range inequalities (e.g., in terms of the dimensions or ranks of the underlying vector spaces) so that the reader can immediately verify which pairs fall inside the claimed ranges.
  2. Notation for the finite field, the symplectic and orthogonal groups, and the oscillator representation should be fixed once in §1 and used consistently; several ad-hoc symbols appear in the abstract and early paragraphs that are not defined until later sections.
  3. The manuscript cites the existence of the oscillator representation as a prerequisite; a brief reference or short appendix recalling the standard construction over finite fields would make the paper self-contained for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript, including the recommendation for minor revision. The report provides a clear summary of the paper's contributions as the first in a series on type I Howe duality over finite fields. No major comments are listed in the provided referee report.

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper asserts an explicit construction of the Howe correspondence (restriction of the oscillator representation) in the two stable ranges. This is presented as a direct construction under the standard assumption that the oscillator representation exists and admits an explicit restriction description when one rank dominates the other. No equations, parameters, or claims reduce by definition or self-citation to the target result itself. The contribution is the construction, not a prediction derived from fitted inputs or prior self-referential theorems. The derivation chain is independent of the output.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.0 · 5651 in / 1036 out tokens · 31205 ms · 2026-05-23T07:07:21.995532+00:00 · methodology

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Reference graph

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