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arxiv: 2606.00679 · v1 · pith:SOZZ65TOnew · submitted 2026-05-30 · 🧮 math.RT · math.CO· math.QA

AHA! RSK

Eugene Stern This is my paper

Pith reviewed 2026-06-28 18:07 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.QA
keywords RSK correspondenceRobinson-Schensted-Knuthdegenerate affine Hecke algebrasymmetric groupJucys-Murphy elementsrectificationjeu de taquinYoung tableaux
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The pith

The RSK correspondence arises exactly as the change of basis between Hecke algebra weight vectors and pairs of Young tableaux labeled by Jucys-Murphy eigenvectors.

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

The paper frames the Robinson-Schensted-Knuth correspondence as a special case of rectification on skew tableaux of staircase shape. Permutations label weight vectors inside a generic module of the degenerate affine Hecke algebra H_n, which is placed inside the regular representation of a larger symmetric group so that external translations act via Jucys-Murphy elements. Rectification is realized by an operator on this representation that performs sequences of exchanges of consecutive entries, squeezing the extra room from the left until only the regular Jucys-Murphy elements of S_n remain. The resulting change of basis maps H_n-weight vectors to S_n-weight vectors that are simultaneous eigenvectors for left and right Jucys-Murphy actions and are therefore labeled by pairs of standard tableaux. The labels match precisely under the RSK bijection.

Core claim

Expressing slides via exchanges of consecutive values inside the tableau lets rectification be modeled by an operator on the regular representation; the explicit change of basis between H_n-weight vectors and S_n-weight vectors, where the latter are eigenvectors of the Jucys-Murphy elements acting both on the left and on the right, produces exactly the RSK correspondence between those labels.

What carries the argument

The rectification operator on the regular representation, realized by sequences of exchanges of consecutive tableau entries, that effects the change of basis from H_n-weight vectors to pairs of S_n Jucys-Murphy eigenvectors.

If this is right

  • The RSK bijection is recovered precisely when the external translations of H_n are reduced to the Jucys-Murphy elements of S_n by the squeezing limit.
  • Rectification of any skew tableau corresponds to applying the same exchange operator inside the regular representation.
  • The initial staircase embedding of a permutation directly supplies the starting skew tableau whose rectification yields the RSK pair.
  • Left and right Jucys-Murphy actions on the regular representation furnish the two tableaux that label each weight vector after the change of basis.

Where Pith is reading between the lines

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

  • The same exchange-operator construction might produce explicit matrix realizations of RSK for larger n without enumerating all permutations.
  • The geometric squeezing picture could be used to deform RSK to other rectification procedures on different posets or to affine versions of the symmetric group.
  • Because the construction is entirely algebraic, it supplies a route to q-deformations or categorifications of RSK inside modules of the (non-degenerate) affine Hecke algebra.

Load-bearing premise

That embedding the weight vectors of the H_n-module into the regular representation of a larger symmetric group correctly encodes the external translations so that rectification becomes a squeezing process leaving only the ordinary Jucys-Murphy elements.

What would settle it

Compute the explicit basis change for all permutations in S_3 or S_4 and verify whether the resulting pairs of standard Young tableaux coincide with the known RSK output for each permutation.

read the original abstract

We give a spectral realization of the Robinson-Schensted-Knuth (RSK) correspondence in terms of the representation theory of the symmetric group $S_n$ and the degenerate affine Hecke algebra (AHA) $H_n$. We view RSK, which builds a pair of standard Young tableaux from a permutation, as a special case of rectification, also known as Jeu de Taquin, which turns skew tableaux into straight ones. In this framing, the initial permutation corresponds to a skew tableau of staircase shape. To interpret this in terms of representation theory, take permutations to label weight vectors in a generic $H_n$-module $V(a_1, \ldots , a_n)$, which is isomorphic to $\mathbb{C}[S_n]$ as an $S_n$-module. Writing permutations as staircases amounts to placing these weight vectors inside the regular representation of a larger symmetric group containing $S_n$; more geometrically, we push $S_n$ to the right toward infinity so its Jucys-Murphy (JM) elements have enough room to represent the external translations of $H_n$. Then, rectification corresponds to squeezing out this extra room from the left, leaving only $S_n$ and its regular JM elements as the limit of the external translations. By expressing slides via sequences of exchanges of consecutive values inside the tableau, we can model rectification by an operator acting on the regular representation. This lets us explicitly write down the change of basis between $H_n$-weight vectors and $S_n$-weight vectors, where the latter are eigenvectors of the JM elements in $S_n$ acting both on the left and on the right, and hence labeled by pairs of standard tableaux. The resulting correspondence between the labels of the weight vectors is exactly RSK.

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 claims a spectral realization of the Robinson-Schensted-Knuth (RSK) correspondence via the representation theory of S_n and the degenerate affine Hecke algebra H_n. Permutations label weight vectors in a generic H_n-module V(a_1,...,a_n) ≅ ℂ[S_n] as an S_n-module. Embedding these into the regular representation of a larger symmetric group (by 'pushing S_n rightward to infinity' so JM elements represent external translations), rectification is modeled as an operator obtained by expressing slides as exchanges of consecutive values. The resulting change of basis between H_n-weight vectors and pairs of S_n JM eigenvectors (labeled by pairs of standard Young tableaux) is asserted to be exactly RSK.

Significance. If the central identification holds, the work would supply a new algebraic model of RSK in terms of Jucys-Murphy elements acting on both sides and a rectification operator on the regular representation, linking the combinatorial jeu de taquin directly to the spectral theory of H_n. The geometric construction of embedding weight vectors and modeling rectification via consecutive exchanges is a potentially useful bridge between combinatorics and degenerate affine Hecke algebra representations.

major comments (2)
  1. [operator definition / rectification modeling] The section defining the rectification operator (via sequences of exchanges of consecutive values): the claim that this operator produces exactly the RSK correspondence is not supported by any explicit matrix or action on basis elements. No computation of the operator for any permutation with n>2 is given, nor is there a direct comparison of the resulting pairs of tableaux against the standard RSK table.
  2. [geometric interpretation / weight vector embedding] The paragraph describing the geometric embedding ('pushing S_n to the right toward infinity' and 'squeezing out extra room'): the identification of the change-of-basis map with RSK rests on this picture without an independent check that the eigenvectors of the limiting JM elements match the insertion and recording tableaux produced by the classical algorithm.
minor comments (2)
  1. [abstract] The abstract introduces 'AHA' only in the title; a parenthetical expansion on first use in the body would improve readability.
  2. [introduction / module setup] Notation for the generic module V(a_1,...,a_n) and the precise definition of the 'external translations' of H_n could be stated more explicitly when first introduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and for recognizing the potential significance of our work. We address each major comment below and will make revisions to the manuscript to provide the requested explicit verifications.

read point-by-point responses

  1. Referee: The section defining the rectification operator (via sequences of exchanges of consecutive values): the claim that this operator produces exactly the RSK correspondence is not supported by any explicit matrix or action on basis elements. No computation of the operator for any permutation with n>2 is given, nor is there a direct comparison of the resulting pairs of tableaux against the standard RSK table.

    Authors: We agree that including explicit computations would make the identification more concrete and verifiable. In the revised manuscript, we will add a new subsection with a full computation of the rectification operator for n=3. This will include the matrix representation of the operator on the 6-dimensional space, its action on each basis element (permutation), and a direct comparison showing that the resulting pairs of standard Young tableaux match those produced by the classical RSK algorithm for each of the 6 permutations. revision: yes

  2. Referee: The paragraph describing the geometric embedding ('pushing S_n to the right toward infinity' and 'squeezing out extra room'): the identification of the change-of-basis map with RSK rests on this picture without an independent check that the eigenvectors of the limiting JM elements match the insertion and recording tableaux produced by the classical algorithm.

    Authors: The geometric picture is meant to provide intuition for why the limiting process corresponds to rectification. However, we acknowledge the need for an independent algebraic check. We will revise the manuscript to include, for the n=3 case, an explicit calculation of the eigenvectors of the limiting JM elements and verify that they are indeed labeled by the insertion and recording tableaux as per RSK. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from module isomorphism and explicit operator modeling to claimed RSK identification

full rationale

The paper frames permutations as weight vectors in a generic H_n-module isomorphic to the regular representation of S_n, embeds them into a larger symmetric group to model external translations via JM elements, defines a rectification operator via sequences of consecutive exchanges, and asserts that the resulting change-of-basis to left/right JM eigenvectors (labeled by pairs of SYT) yields exactly RSK. No quoted equation or step reduces the target correspondence to a fitted parameter, self-definition, or unverified self-citation; the construction is presented as self-contained against the representation-theoretic inputs, with the RSK claim as a derived consequence rather than an input. This matches the default expectation of an independent realization.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard facts about S_n-modules and H_n-modules together with the paper-specific modeling of rectification as an operator via exchanges; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption A generic H_n-module V(a1,...,an) is isomorphic to C[S_n] as an S_n-module
    This isomorphism is invoked to label weight vectors with permutations.
  • ad hoc to paper Rectification corresponds to an operator on the regular representation obtained by modeling slides as exchanges of consecutive values
    This modeling step is required to produce the change of basis that yields RSK.

pith-pipeline@v0.9.1-grok · 5846 in / 1444 out tokens · 35482 ms · 2026-06-28T18:07:29.545136+00:00 · methodology

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Reference graph

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