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arxiv: 2506.22983 · v2 · submitted 2025-06-28 · 🧮 math.RT

Howe duality over finite fields II: Explicit stable computation

Sophie Kriz This is my paper

Pith reviewed 2026-05-19 08:22 UTC · model grok-4.3

classification 🧮 math.RT
keywords type I Howe dualityfinite fieldseta correspondencezeta correspondencecharacter parametrizationstable rangesfinite groups of Lie typeirreducible representations
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The pith

The eta and zeta correspondences of type I Howe duality over finite fields are explicitly described using the parametrization of irreducible characters of finite groups of Lie type in the stable ranges.

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

The paper works to translate the eta and zeta correspondences, built in the first part of the series, into the language of the parametrization of irreducible characters of finite groups of Lie type. This translation is carried out specifically in the two stable ranges. A sympathetic reader would value the result because it turns the known non-zero multiplicities from earlier results into a direct matching of character parameters, making the duality more explicit and potentially easier to apply in calculations.

Core claim

In the two stable ranges, the eta and zeta correspondences are explicitly described using the parametrization of irreducible characters of finite groups of Lie type. This pins down which pairs of representations participate in the stable eta and zeta correspondences among those already known to appear with positive multiplicity in the type I Howe duality by earlier results.

What carries the argument

The translation of the eta and zeta correspondences into the parametrization of irreducible characters of finite groups of Lie type.

If this is right

  • The pairs of representations in the stable Howe duality can now be listed by their character parameters rather than abstractly.
  • The explicit form confirms the identification of the correspondences within the ranges where non-zero multiplicities are established by earlier work.
  • Further computations of the duality can reference the standard character tables and parametrizations for these groups.

Where Pith is reading between the lines

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

  • This explicit description provides a basis for attempting similar translations in non-stable ranges where the multiplicity behavior is less understood.
  • Examining the specific mappings between parameters may uncover combinatorial rules governing the duality.

Load-bearing premise

The constructions of the eta and zeta correspondences from the first paper remain valid and translate directly into the parametrization of irreducible characters in the stable ranges without further modification.

What would settle it

Observe a specific pair of irreducible representations in a stable range whose Howe duality multiplicity is positive yet whose parameters in the character parametrization do not satisfy the explicit eta or zeta relation as described.

read the original abstract

In this second paper of a series dedicated to type I Howe duality for finite fields, we explicitly describe the eta and zeta correspondences constructed in the first paper in terms of G. Lusztig's parametrization of the irreducible characters of finite groups of Lie type in the two so-called stable ranges. This identifies the stable eta and zeta correspondences among the pairs of irreducible representations whose occurence with non-zero multiplicity in the type I Howe duality correspondence was proved by S.-Y. Pan.

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 explicitly describes the eta and zeta correspondences constructed in the first paper of the series in terms of G. Lusztig's parametrization of irreducible characters of finite groups of Lie type, restricted to the two stable ranges. It identifies these stable correspondences with the pairs of irreducible representations that occur with non-zero multiplicity in the type I Howe duality, as previously shown by S.-Y. Pan.

Significance. If the identification is correct, the result supplies an explicit, computable form for the stable part of the Howe correspondence over finite fields by leveraging the standard Lusztig parametrization. This advances the program of the series by converting the abstract correspondences of part I into concrete pairs of parameters, building directly on Pan's multiplicity theorem without introducing new free parameters or ad-hoc constructions.

minor comments (3)
  1. The introduction would benefit from a short paragraph recalling the precise definitions of the eta and zeta correspondences from part I, to make the translation self-contained for readers who have not recently consulted the first paper.
  2. In the statement of the main identification (likely Theorem 3.1 or 4.1), the precise mapping between Lusztig parameters and the stable-range representations could be written out as an explicit bijection rather than described narratively.
  3. A brief remark on the range of finite fields for which the stable-range hypothesis holds would clarify the scope, even if it follows from prior results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the main contribution of the paper.

read point-by-point responses

  1. Referee: The paper explicitly describes the eta and zeta correspondences constructed in the first paper of the series in terms of G. Lusztig's parametrization of irreducible characters of finite groups of Lie type, restricted to the two stable ranges. It identifies these stable correspondences with the pairs of irreducible representations that occur with non-zero multiplicity in the type I Howe duality, as previously shown by S.-Y. Pan.

    Authors: We appreciate the referee's concise and accurate summary of the results. The explicit identification in the stable ranges is the central point of the paper, obtained by combining the abstract correspondences from part I with Lusztig's parametrization and Pan's multiplicity theorem. No new free parameters are introduced. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper performs an explicit re-labeling of the eta and zeta correspondences (whose existence and basic properties are taken from the author's prior Part I) into the external Lusztig parametrization of irreducible characters of finite groups of Lie type, restricted to the stable ranges already known to be multiplicity-free by Pan's independent theorem. No derivation step equates a claimed prediction or first-principles result to its own inputs by construction, no fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the same authors' prior work as an external fact. The central identification therefore rests on external benchmarks (Lusztig, Pan) plus the prior construction, rendering the derivation self-contained rather than circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the eta/zeta constructions from the first paper in the series and on Lusztig's established parametrization of characters for groups of Lie type; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math Lusztig's parametrization correctly labels all irreducible characters of the relevant finite groups of Lie type
    Invoked when translating the correspondences into explicit form in the stable ranges.
  • domain assumption The multiplicity results of S.-Y. Pan hold for the pairs under consideration
    Used to identify which pairs correspond to the stable eta and zeta maps.

pith-pipeline@v0.9.0 · 5589 in / 1423 out tokens · 47894 ms · 2026-05-19T08:22:37.053439+00:00 · methodology

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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