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Cyclic sieving, promotion, and representation theory

Brendon Rhoades

arXiv:1005.2568v1math.COmath.RT

TL;DR

The paper addresses the lack of an enumerative proof for cyclic sieving on rectangular tableaux under jeu-de-taquin promotion. It takes an algebraic approach and establishes cyclic-sieving and related fixed-point results, with a modulus caveat for column-strict tableaux.

  • Problem

    No enumerative proof is known for the cyclic-sieving results on rectangular tableaux described here.

    Source: §1 · p. 4

  • Method

    The paper approaches cyclic sieving through algebraic structure, using epimorphisms to study representations and the vanishing properties of Kazhdan–Lusztig immanants.

    Source: §1 · p. 4, §5 · p. 36

  • Results

    For rectangular standard tableaux, promotion exhibits CSP with the q-hook formula and a q-shifted Schur specialization; column-strict fixed-point counts equal the modulus of Kλ,α(ζ^m).

    Source: §1 · pp. 2–4

  • Takeaways & Limitations

    The paper treats cyclic-sieving results as concealing algebraic structure that can be exploited in proofs.

    Source: §1 · p. 4

    The column-strict result requires rectangular shape, and possible negative Kostka–Foulkes evaluations make it a cyclic-sieving result only up to modulus.

    Source: §1 · p. 4, §5 · p. 38

Abstract

from arXiv · show

We prove a collection of conjectures of D. White \cite{WComm}, as well as some related conjectures of Abuzzahab-Korson-Li-Meyer \cite{AKLM} and of Reiner and White \cite{ReinerComm}, \cite{WComm}, regarding the cyclic sieving phenomenon of Reiner, Stanton, and White \cite{RSWCSP} as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of $\mathbb{C}[x_{11}, ..., x_{nn}]$ due to Skandera \cite{SkanNNDCB}. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups and noncrossing partitions.

1. Introduction

The introduction establishes cyclic sieving as a way to encode fixed-point counts through polynomial evaluations, then states results for promotion on rectangular tableaux. It motivates a representation-theoretic proof strategy and previews extensions to fixed-content tableaux and other combinatorial actions.

1. Introduction

  • Cyclic sieving associates a polynomial to a cyclic action so its root-of-unity evaluations encode fixed-point counts.Those fixed-point counts determine the cycle structure of the action; natural polynomial deformations of counting formulas are of particular interest.Source: §1 · pp. 1–2
  • For rectangular shape λ, promotion on standard tableaux exhibits cyclic sieving with the q-analogue of the hook length formula.The introduction also identifies this polynomial, up to a power of q, with a fake degree polynomial for the λ-isotypic component of the symmetric-group coinvariant algebra.Source: §1 · pp. 2–3
  • For rectangular shape λ, promotion on column strict tableaux with entries bounded by k exhibits cyclic sieving with a q-shift of the principal specialization of sλ.The associated polynomial is also related to the q-hook-content formula and to a generating function for plane partitions in a box.Source: §1 · p. 3
  • For rectangular λ and cyclically symmetric content α of length k, promotion fixed-point counts equal |Kλ,α(ζ^m)| for every power m.This fixed-content result is a weight-space refinement of the column-strict result.Source: §1 · p. 4
  • The fixed-content theorem is cyclic sieving only up to modulus: Kostka-Foulkes evaluations at roots of unity can be negative.The introduction also presents the standard-tableau result as a special case of the fixed-content result, up to modulus.Source: §1 · p. 4
  • The authors turn to representation theory because no enumerative proof is known for the rectangular-tableau cyclic-sieving results, unlike the subset and multiset cases.They state that their tableau results also imply the subset and multiset results.Source: §1 · pp. 2, 4
  • The proof strategy represents the cyclic action on a vector space so fixed-point counts become character traces, then relates those character values to root-of-unity evaluations.The authors note that this can require more than direct conjugacy, including Springer’s theory of regular elements for the standard-tableau result.Source: §1 · pp. 5–6
  • The introduction previews fixed-point results for dihedral actions and applications involving handshake patterns and the reflection group Bn.The paper also closes with open questions.Source: §1 · p. 6

2. Tableaux and Representation Theory Background

This section develops the tableau and jeu-de-taquin promotion framework alongside the representation-theoretic tools used to study it, including Kazhdan–Lusztig theory, RSK, and evacuation. It connects RSK/Knuth classes with Kazhdan–Lusztig cells and sets up tableau-indexed representations and operator models for evacuation and promotion on rectangular tableaux.

Tableaux and promotion

  • Promotion removes entries equal to the bound, slides the resulting dots northwest, then increments entries and fills holes with 1; it is bijective and cyclically rotates content.The promotion algorithm’s result depends on the chosen upper bound for the tableau entries.Source: §2 · pp. 8–9

Kazhdan–Lusztig theory

  • The Kazhdan–Lusztig basis specializes from H_n(q) to C[S_n] and provides a basis in which the regular representation’s decomposition into irreducibles becomes visible.Its defining properties include invariance under an involution, Bruhat compatibility, and a degree bound on its polynomial coefficients.Source: §2 · pp. 10, 12

RSK and KL cells

  • RSK bijects permutations with pairs of standard tableaux of the same shape, while the KL preorders partition permutations into left, right, and two-sided cells.These cells are identified by an explicit combinatorial algorithm.Source: §2 · p. 13

Evacuation

  • Evacuation is a well-defined involution on tableaux that reverses content; its RSK interactions send permutations involving the long element to tableau pairs transformed by evacuation and conjugation.For w mapping to (P, Q), the double product with the long element maps to (e(P), e(Q)).Source: §2 · p. 14

Descent sets

  • RSK carries left and right permutation descents to descents of P and Q, while multiplication by the long element transforms tableau pairs through evacuation and conjugation; rectangular cases motivate extended descents.The section introduces extended descent sets for rectangular tableaux as useful in proving later representation-theoretic results.Source: §2 · p. 14

Tableaux and Kazhdan–Lusztig cells

  • RSK left and right Knuth classes correspond to left and right Kazhdan–Lusztig cells, while equal-shape permutations correspond to two-sided cells ordered by dominance.The left and right cell orders induce the same partial order on standard tableaux.Source: §2 · pp. 14–15

Change of label

  • Change of label makes the symmetrized KL µ-value between tableaux of the same shape independent of the auxiliary tableau used to define it.The value is defined as the common value of µ[(P,T),(Q,T)] and µ[(T,P),(T,Q)] for any same-shape T.Source: §2 · p. 15

KL cellular representations

  • The KL cellular module Sλ is the irreducible symmetric-group representation indexed by λ, with basis naturally bijective to SYT(λ).Its Coxeter-generator action is determined by the symmetrized KL µ-function and tableau descent sets, and does not depend on the auxiliary tableau used in its construction.Source: §2 · p. 16

Evacuation and promotion

  • The long element represents evacuation up to sign in the KL cellular representation, and the long cycle is to model promotion for rectangular partitions.The sign is determined by the Murnaghan–Nakayama rule; the paper says it resolves the sign by direct KL-basis manipulation in the next section.Source: §2 · pp. 16–17

3. Promotion on Standard Tableaux

This section studies promotion on standard tableaux through its relationship to Kazhdan–Lusztig theory, including preservation of the μ function and its connection to the long cycle, and establishes cyclic sieving for rectangular shapes with consequences for μ-coefficients. It also describes promotion’s action on extended descent sets and notes limitations beyond rectangles, where the analogous q-hook-length formula need not give cyclic sieving.

Promotion and the μ function

  • Promotion preserves μ[P,Q] when P and Q both have rectangular shape or both have a rectangle with its outer corner removed.For other shapes this need not hold; the shapes (3,1) and (2,1,1) give counterexamples.Source: §3 · p. 18

Limitations beyond rectangles

  • For nonrectangular shapes, promotion need not preserve μ, and no extended descent set agreeing with ordinary descents is generally known on which it acts with order n.For shape (3,2), promotion has order 6 although the shape has 5 boxes.Source: §3 · pp. 18, 20

Extended descent sets

  • On rectangular tableaux, promotion cyclically rotates the extended descent set, which extends the ordinary descent set by possibly including n.The extended set’s n-membership is determined by whether n lies north of the jeu-de-taquin hole after deleting 1.Source: §3 · p. 20

Promotion in the KL representation

  • For a rectangular shape λ = ba with ab = n, the KL image of the long cycle is (−1)^(a−1) times the permutation matrix for promotion.The scalar is determined using the coefficient computed in Lemma 3.4.Source: §3 · pp. 23–24

Promotion order

  • Every rectangular standard tableau is fixed by j^n, so promotion’s order divides n but need not equal n.Single-row and single-column shapes have promotion order 1; (2,2) is another counterexample to order n.Source: §3 · pp. 24–25

Staircase shapes

  • For staircase shapes ν = (k, k −1, …, 1), j^2n fixes every tableau and j^|ν| acts by transposition, but lower positive powers need not preserve μ.For ν = (3,2,1), no power j^d for 1 ≤ d ≤ 5 preserves μ.Source: §3 · p. 25

3. Promotion on Standard Tableaux

  • For rectangular λ, μ[P,P′]=μ[Q,Q′] whenever P,Q lie in one promotion orbit and P′,Q′ lie in another whose order is relatively prime to the first.In the λ=(2,2,2) example, the common μ-value for the specified pairs of column-reading words is 1.Source: §3 · p. 26
  • For λ=(2,2,2), the q-hook polynomial evaluates at 1, ζ, ζ², ζ³, ζ⁴, ζ⁵ to 5, 0, 2, 3, 2, 0, matching the corresponding promotion fixed-point counts.The promotion action has one orbit of size 2 and one of size 3.Source: §3 · pp. 26–27
  • For rectangular λ, jeu-de-taquin promotion on SYT(λ) exhibits cyclic sieving with the q-hook-length polynomial.The proof relates the character value on powers of the long cycle to the trace of a sign-adjusted power of the promotion matrix, hence to promotion fixed-point counts.Source: §3 · pp. 26–27
  • For λ=(3,3,1), promotion has orbits of sizes 3, 5, and 13, and the q-hook-length polynomial is nonreal at a primitive 195th root, so it fails to count fixed points.The promotion order is 195, while the one-step fixed-point count is zero.Source: §3 · p. 26

4. A Construction of the Finite Dimensional Irreducible GLk(C) Representations

This section constructs finite-dimensional GLₖ(ℂ) modules from Kazhdan–Lusztig immanants, identifying their irreducible type and tableau-indexed weight spaces. It illustrates the construction in the (2,2), k=3 example by enumerating the RST set and forming a basis element from a 3412-KL immanant.

4. A Construction of the Finite Dimensional Irreducible GLk(C) Representations

  • For fixed insertion tableau T, a quotient of the KL-immanant span carries the irreducible Sn representation of shape λ, with basis indexed by standard tableaux.The representation does not depend on the choice of T.Source: §4 · p. 30
  • A KL immanant Imm(T,U)(xα,1n) vanishes exactly when U is not α-semistandardizable; the nonzero immanants across these compositions and tableaux are linearly independent.The criterion connects immanant nonvanishing to tableau semistandardizability.Source: §4 · pp. 31–32
  • The GLk(C) module is formed as a quotient with basis given by immanant images indexed by compositions α and α-semistandardizable tableaux.Each basis element Iα(U′) corresponds to a row-strict tableau U′ whose standardization is U.Source: §4 · p. 34
  • For shape λ, VT,k is dual to the irreducible finite-dimensional GLk(C) representation of highest weight λ′, with weight −α spanned by tableaux of content α.The basis is indexed by row-strict tableaux of shape λ and content α.Source: §4 · p. 34
  • In the example with k = 3 and λ = (2, 2), the GL3(C)-stable space has dimension 86, while its quotient module has dimension 6.The quotient dimension equals the number of row-strict tableaux of shape (2, 2) with entries at most 3.Source: §4 · pp. 33, 35
  • RST((2,2),3) is identified as a six-element set of tableaux.The set is listed explicitly in the passage.Source: §4 · p. 35
  • A basis element is constructed by applying the 3412-KL immanant, then taking its homomorphic image in the quotient V′.The immanant is x13x24x21x32 − x14x23x21x32 − x13x24x22x31 + x14x23x22x31.Source: §4 · p. 35

5. Promotion on Column Strict Tableaux

For rectangular shapes, the paper proves a cyclic sieving result for column-strict tableaux under jeu-de-taquin promotion. A representation-theoretic construction relates promotion to long-cycle actions and extends the standard-tableau result to semistandard tableaux.

5. Promotion on Column Strict Tableaux

  • The maps π_α transfer information from standard to semistandard tableaux by relating long-cycle actions to promotion.The maps send immanants to zero unless the tableau is α-semistandardizable; the resulting images furnish the relevant basis vectors.Source: §5 · pp. 36–37
  • For rectangular λ, promotion has order k on row-strict tableaux of content length k, except for the stated single-column and single-row cases.The exceptions occur when λ is a single column or row, respectively, and k equals |λ|.Source: §5 · p. 38
  • The rectangular-shape hypothesis is necessary for the column-strict sieving result; for arbitrary shapes, promotion’s order is unknown and changing the group order does not suffice.The failure persists when the cyclic group is chosen to have the promotion order and X(q) is the principal specialization of s_λ.Source: §5 · p. 38
  • For rectangular λ, the long-cycle action on the constructed representation yields the cyclic sieving result for column-strict tableaux under promotion.The polynomial is a specialization of a Schur function, evaluated at roots of unity.Source: §5 · pp. 35, 39
  • The paper derives Reiner–Stanton–White’s introduction results as corollaries by specializing the column-strict theorem to a single column or row.The proofs interchange n and k and use the corresponding single-column or single-row shape.Source: §5 · p. 39

6. Promotion on Column Strict Tableaux with Fixed Content

For rectangular shapes and cyclically symmetric content, the paper studies powers of promotion on fixed-content column-strict tableaux through Kostka–Foulkes polynomials. The resulting fixed-point relation is a near-CSP, with root-of-unity evaluations controlled in modulus by ribbon tableaux.

6. Promotion on Column Strict Tableaux with Fixed Content

  • For rectangular λ and cyclically symmetric α, powers of promotion preserve the fixed-content tableau set and yield a near-CSP involving Kostka–Foulkes polynomials.This is a weight-space refinement of the preceding results.Source: §6 · p. 39
  • The Kostka–Foulkes root-of-unity evaluation may be negative, so it does not give a CSP directly, while the fixed-point relation holds up to modulus.The polynomial is a q-analogue of the Kostka number that counts column-strict tableaux of the given shape and content.Source: §6 · p. 40
  • At a root of unity of order d, the Kostka–Foulkes evaluation vanishes unless every part multiplicity of α is divisible by d; otherwise its modulus counts column-strict d-ribbon tableaux.The ribbon-tableau content is formed from the part multiplicities divided by d.Source: §6 · p. 41
  • The representation-theoretic argument uses weight spaces: the long cycle maps the space of content β to that of cyclically shifted content, while its powers yield character evaluations.The proof converts these evaluations into identities for the content compositions fixed by cyclic shifts.Source: §6 · p. 43
  • A genuine CSP would require a q-shift of the Kostka–Foulkes polynomial whose shift accounts for the root-of-unity sign, but the paper does not identify such a shift.The authors state this as a desirable strengthening of their result.Source: §6 · pp. 43–44

7. Dihedral Actions

The section studies the dihedral action generated by promotion and evacuation on rectangular tableaux, modeling it through symmetric-group actions. It derives fixed-point results while noting that a polynomial interpretation for dihedral fixed-point counts remains unknown.

7. Dihedral Actions

  • A dihedral sieving interpretation of these fixed-point counts remains an open question for the rectangular tableau actions considered.The paper hopes this analysis will motivate study of such a phenomenon.Source: §7 · p. 44
  • Evacuation and promotion generate a dihedral group of order 2k on the rectangular tableau sets, with evacuation conjugating promotion to its inverse.The promotion operator has order k and evacuation has order 2.Source: §7 · p. 44
  • The dihedral tableau action is modeled by the long cycle and long element in symmetric groups, enabling fixed-point counts for evacuation and evacuation-promotion to be expressed as character evaluations.For standard tableaux, the section identifies these fixed-point counts as character evaluations; promotion powers are already counted by q-hook evaluations.Source: §7 · pp. 44, 46
  • Reflection cycle types depend on parity: for odd n the reflections form one conjugacy class, while even n has two classes fixing zero or two vertices.The Murnaghan–Nakayama rule is used to determine the relevant character signs for the rectangular shapes.Source: §7 · pp. 44–46
  • The fixed-point formulas for evacuation and evacuation-promotion distinguish odd and even k and depend on the rectangle’s parity, with Schur-function evaluations in the stated cases.The formulas are given for rectangular tableaux, including column-strict tableaux.Source: §7 · p. 48
  • For column-strict tableaux, promotion’s symmetric-group model carries a sign that can vary by weight space, so fixed points need not equal a character value up to sign for every group element.The section attributes this qualification to the long cycle’s weight-dependent sign.Source: §7 · p. 49

8. Applications to other Combinatorial Actions

The paper extends its cyclic-sieving results to reduced words, handshake patterns, and noncrossing partitions, and analyzes reflection-fixed points under related dihedral actions.

Applications to other Combinatorial Actions

  • Reduced words for the long element of B_n exhibit CSP under cyclic rotation, with q-hook-length polynomial f_nn(q) and cyclic group Z/n^2Z.The result is transported through bijections from standard tableaux to shifted tableaux and then to reduced words.Source: §8 · p. 49
  • Handshake patterns under rotation and noncrossing partitions under Kreweras complementation both exhibit CSP with the q-Catalan polynomial.Bijections to two-row standard tableaux carry these actions to promotion; the result also gives new proofs of results of White and Heitsch.Source: §8 · pp. 50–51
  • Handshake patterns and noncrossing partitions carry dihedral actions whose fixed-point counts for r and rs are given by formulas.On handshake patterns, r corresponds to evacuation; the actions give an explicit realization of the dihedral actions investigated earlier.Source: §8 · pp. 50, 52

9. Open Problems

The section records bijective consequences of the rectangular-tableau results and poses questions about extending promotion results to arbitrary partitions.

Open Problems

  • Corollaries 9.1–9.3 give bijections for promotion-fixed standard and column-strict tableaux, including a correspondence with d-ribbon tableaux.The stated conditions include d|n or d|k, and the content-restricted case requires c_d·α=α.Source: §9 · p. 52
  • Problem 9.4 asks for explicit bijections between the pairs of sets in those three corollaries.The preceding results establish correspondences but leave explicit constructions as an open problem.Source: §9 · pp. 52–53
  • For arbitrary partitions, the section asks for promotion's order and cycle structure and for a polynomial giving a CSP, beyond the rectangular setting.The section notes that promotion's nth power need not fix every standard tableau for nonrectangular shapes.Source: §9 · p. 53
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