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arxiv: 2605.23609 · v1 · pith:TQZPW5J7new · submitted 2026-05-22 · 🧮 math.RT · math.QA

On reciprocal characters and the quantum affine Schur-Weyl duality

Maxim Gurevich , Angelina Vargulevich This is my paper

Pith reviewed 2026-05-25 02:47 UTC · model grok-4.3

classification 🧮 math.RT math.QA MSC 17B3720C08
keywords reciprocal characteraffine Hecke algebraquantum affine Schur-Weyl dualityq-characterdominant monomialparabolic restrictiontableau countingLanglands parameterization
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The pith

The reciprocal character from affine Hecke algebras equals the dominant q-character of quantum affine modules under Schur-Weyl duality.

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

The paper defines the reciprocal character of modules over GL_n-type affine Hecke algebras via multiplicities in parabolic restriction. It proves this character equals the dominant part of the Frenkel-Reshetikhin q-character through quantum affine Schur-Weyl duality. The result supplies a type A case of the expectation that dominant monomials coordinate the monomial basis in Lusztig's framework for quantum groups. Under the affine Hecke categorification, the reciprocal character specializes at q=1 to the coordinate map on that basis. Consequently, multiplicities of dominant q-characters are read off as transition coefficients between monomial and canonical or PBW bases, with explicit formulas obtained by counting tableaux.

Core claim

We identify the dominant part of the Frenkel-Reshetikhin q-character with a natural invariant arising from the Langlands/Zelevinsky parameterization for affine Hecke algebras. We introduce the reciprocal character of a module over a GL_n-type affine Hecke algebra, defined in terms of multiplicities within parabolic restriction. The main theorem claims that the reciprocal character matches, under quantum affine Schur-Weyl duality, with the dominant q-character for finite-dimensional modules over quantum affine algebras. This result gives a type A realization of the Nakajima expectation that the dominant monomials in the q-character should play the role of monomial-basis coordinates in Lusztig

What carries the argument

The reciprocal character, defined via multiplicities in parabolic restriction of modules over GL_n-type affine Hecke algebras and matched to dominant q-characters by quantum affine Schur-Weyl duality.

If this is right

  • Dominant q-character multiplicities for simple modules equal the transition coefficients between monomial and canonical bases.
  • Dominant q-character multiplicities for standard modules equal the transition coefficients between monomial and PBW bases.
  • Explicit tableau-counting formulas compute the dominant multiplicities, or equivalently the reciprocal characters of standard modules.
  • The Langlands/Zelevinsky parameterization for affine Hecke algebras supplies a type A model for the dominant monomials in q-characters.

Where Pith is reading between the lines

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

  • Tableau formulas may yield practical algorithms for computing q-character multiplicities without direct reference to quantum affine modules.
  • The type A identification raises the question of analogous reciprocal characters in other Dynkin types via suitable Hecke or quiver varieties.
  • The basis-transition description suggests that positivity properties of q-characters could be proved combinatorially from Hecke algebra data.

Load-bearing premise

The affine Hecke categorification of U_q(sl_infty)^+ holds, so the reciprocal character equals the q=1 specialization of the coordinate map on the monomial basis.

What would settle it

A concrete standard module over an affine Hecke algebra whose reciprocal character, computed from parabolic restriction multiplicities, differs from the dominant q-character of the corresponding quantum affine module.

read the original abstract

We identify the dominant part of the Frenkel-Reshetikhin $q$-character with a natural invariant arising from the Langlands/Zelevinsky parameterization for affine Hecke algebras. We introduce the reciprocal character of a module over a $GL_n$-type affine Hecke algebra, defined in terms of multiplicities within parabolic restriction. The main theorem claims that the reciprocal character matches, under quantum affine Schur--Weyl duality, with the dominant $q$-character for finite-dimensional modules over quantum affine algebras. This result gives a type $A$ realization of the Nakajima expectation that the dominant monomials in the $q$-character should play the role of monomial-basis coordinates in Lusztig's framework for finite quantum groups. Indeed, under the affine Hecke categorification of $U_q(\mathfrak{sl}_\infty)^+$, we prove that the reciprocal character is the specialization at $q=1$ of the coordinate map attached to a monomial basis. As a consequence, dominant $q$-character multiplicities for simple (or standard) modules are described by transition coefficients between monomial and canonical (or PBW) bases. Our methods rely on the development of explicit tableau-counting formulas for such dominant multiplicities, or equivalently for the reciprocal characters of standard modules over affine Hecke algebras.

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 / 2 minor

Summary. The paper introduces the reciprocal character of a module over a GL_n-type affine Hecke algebra, defined via multiplicities in parabolic restriction with respect to the Langlands-Zelevinsky parameterization. The central claim is that, under quantum affine Schur-Weyl duality, this reciprocal character coincides with the dominant part of the Frenkel-Reshetikhin q-character of the corresponding finite-dimensional module over a quantum affine algebra. Invoking the affine Hecke categorification of U_q(sl_∞)^+, the authors prove that the reciprocal character equals the q=1 specialization of the coordinate map attached to a monomial basis; as a consequence, dominant q-character multiplicities for simple or standard modules are given by the transition coefficients between the monomial basis and the canonical (or PBW) basis. Explicit tableau-counting formulas are derived for the reciprocal characters of standard modules (equivalently, for the dominant multiplicities).

Significance. If the identification is valid, the work supplies a concrete type-A realization of Nakajima's expectation that dominant monomials in q-characters function as monomial-basis coordinates in Lusztig's framework. It furnishes an explicit bridge between invariants of affine Hecke algebras and the representation theory of quantum affine algebras, together with combinatorial tableau formulas that yield falsifiable predictions for multiplicities. The explicit development of these counting formulas constitutes a verifiable strength of the manuscript.

major comments (2)
  1. [paragraph on affine Hecke categorification and coordinate-map property] The paragraph invoking the affine Hecke categorification of U_q(sl_∞)^+ asserts that the reciprocal character is the q=1 specialization of the monomial-basis coordinate map, but supplies no explicit verification that the categorification sends the standard modules (under the Langlands-Zelevinsky parameterization) to the expected PBW or canonical basis elements. Without this check, the subsequent claim that dominant multiplicities are transition coefficients does not follow.
  2. [section developing tableau-counting formulas] The derivation of the tableau-counting formulas for dominant multiplicities (equivalently, for reciprocal characters of standard modules) is presented as a direct consequence of the coordinate-map identification; if the categorification map fails to preserve the relevant parameterization, these formulas do not correctly compute the q-character multiplicities.
minor comments (2)
  1. The abstract refers to 'the main theorem' without a theorem number; the manuscript should label the principal result explicitly (e.g., Theorem 4.5) for reference.
  2. Notation for the reciprocal character (e.g., the precise definition via parabolic restriction multiplicities) should be introduced with a displayed equation rather than inline text to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the exposition of the categorification argument can be strengthened. We respond to each major comment below.

read point-by-point responses

  1. Referee: The paragraph invoking the affine Hecke categorification of U_q(sl_∞)^+ asserts that the reciprocal character is the q=1 specialization of the monomial-basis coordinate map, but supplies no explicit verification that the categorification sends the standard modules (under the Langlands-Zelevinsky parameterization) to the expected PBW or canonical basis elements. Without this check, the subsequent claim that dominant multiplicities are transition coefficients does not follow.

    Authors: We agree that an explicit verification of the correspondence between Langlands-Zelevinsky standard modules and the PBW basis elements under the categorification would make the argument clearer. In the revised manuscript we will insert a short subsection (expanding the paragraph in Section 3) that recalls the known functorial properties of the affine Hecke categorification of U_q(sl_∞)^+ and directly checks that the standard modules map to the expected PBW generators; this will confirm that the reciprocal character is indeed the q=1 coordinate map and that the transition-coefficient claim follows. revision: yes

  2. Referee: The derivation of the tableau-counting formulas for dominant multiplicities (equivalently, for reciprocal characters of standard modules) is presented as a direct consequence of the coordinate-map identification; if the categorification map fails to preserve the relevant parameterization, these formulas do not correctly compute the q-character multiplicities.

    Authors: The tableau-counting formulas themselves are derived combinatorially in Section 5 from the affine Hecke side alone. Their identification with dominant q-character multiplicities rests on the main theorem. To address the referee’s conditional concern we will add a remark after the formulas that cross-references the new verification subsection (added in response to the first comment) and notes that the Langlands-Zelevinsky parameterization is preserved by the categorification, thereby ensuring the formulas compute the correct multiplicities. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external duality and supplies independent tableau formulas

full rationale

The paper defines the reciprocal character independently from parabolic restriction multiplicities on affine Hecke algebras. The main theorem equates this to the dominant q-character under the quantum affine Schur-Weyl duality (an external, standard result). The affine Hecke categorification of U_q(sl_∞)^+ is invoked only for the interpretive consequence that the reciprocal character specializes to a monomial-basis coordinate map, after which transition coefficients describe multiplicities; the paper develops explicit tableau-counting formulas for these multiplicities as new, self-contained content. No step reduces a claimed prediction or identification to a fitted input, self-definition, or load-bearing self-citation by construction. The categorification functions as external input rather than the derivation collapsing to its own assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper introduces one new entity (reciprocal character) and relies on standard domain assumptions about duality and categorification; no free parameters are visible in the abstract.

axioms (2)
  • domain assumption Existence and properties of quantum affine Schur-Weyl duality
    Invoked to equate the two characters in the main theorem.
  • domain assumption Existence of the affine Hecke categorification of U_q(sl_∞)^+
    Required for the specialization statement and the coordinate-map interpretation.
invented entities (1)
  • reciprocal character no independent evidence
    purpose: Invariant defined via multiplicities in parabolic restriction for affine Hecke algebra modules
    Newly introduced to match the dominant q-character.

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