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Promotion and evacuation on standard Young tableaux of rectangle and staircase shape

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promotion-and-evacuation-on-standard-young-tableaux-of-rectangle-andRepresentation Theorymath.GRmath.RTposed by Steven Pon, Qiang Wangrecorded: open · 1 machine check, unexamined

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Statement

For the case k>5, Corollary 4.19 gives a necessary condition for what kind of cycles can appear in the promotion action on SYT(sck)SYT(sc_{k}) . We do not know if this condition is sufficient.

Context

Candidate 1 of the open problems stated in "Promotion and evacuation on standard Young tableaux of rectangle and staircase shape", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: Reconstructed question: for k>5k>5, let sck=(k,k−1,…,1)sc_k=(k,k-1,\ldots,1), n=∣sck∣=k(k+1)/2n=|sc_k|=k(k+1)/2, and let ∂\partial be Schützenberger promotion on SYT(sck)SYT(sc_k), whose order is N=k(k+1)=2nN=k(k+1)=2n. Pon--Wang Corollary 4.19 says that if a promotion cycle has length CC, then

    C∣NandC∤n.C\mid N\quad\text{and}\quad C\nmid n.

    The sufficiency question asks whether every such CC occurs as a promotion-cycle length.

    Result: The condition is not sufficient.

    Take k=6k=6. Then N=42N=42, n=21n=21, and C=2C=2 satisfies

    2∣42,2∤21.2\mid 42,\qquad 2\nmid 21.

    I show that no promotion 2-cycle exists on SYT(sc6)SYT(sc_6).

    Use the Edelman--Greene correspondence between SYT(sck)SYT(sc_k) and reduced words

    (a1,…,an)(a_1,\ldots,a_n)

    for the longest permutation w0∈Sk+1w_0\in S_{k+1}, with simple generators si=(i,i+1)s_i=(i,i+1). Under this correspondence promotion is intertwined, up to convention/inverse, with

    ρ(a1,…,an)=(a2,…,an,k+1−a1).\rho(a_1,\ldots,a_n)=(a_2,\ldots,a_n,k+1-a_1).

    For k=6k=6, we have k+1=7k+1=7, n=21n=21. If a tableau had promotion period dividing 22, the corresponding reduced word would satisfy ρ2(a)=a\rho^2(a)=a. But

    ρ2(a1,…,a21)=(a3,…,a21,7−a1,7−a2),\rho^2(a_1,\ldots,a_{21}) =(a_3,\ldots,a_{21},7-a_1,7-a_2),

    so equality forces

    ai=ai+2(1≤i≤19),a_i=a_{i+2}\quad(1\le i\le 19),

    and hence all odd positions equal some aa, all even positions equal some bb, with

    b=7−a,a=7−b.b=7-a,\qquad a=7-b.

    Thus the word uses only the two generators sas_a and s7−as_{7-a}. Its product therefore lies in the proper parabolic subgroup generated by these two adjacent transpositions. It cannot equal w0∈S7w_0\in S_7, whose reduced expressions require all six simple generators s1,…,s6s_1,\ldots,s_6.

    Contradiction. Hence no tableau in SYT(sc6)SYT(sc_6) is fixed by ∂2\partial^2, so no 2-cycle occurs, despite C=2C=2 satisfying Pon--Wang’s necessary condition.

    Audit: the counterexample has k=6>5k=6>5, uses exactly the stated necessary condition, and disproves sufficiency without adding hypotheses.

    Citation: The counterexample above is the resolution. Background used: P. Edelman and C. Greene, “Balanced tableaux,” Adv. Math. 63 (1987), 42–99; S. Pon and Q. Wang, “Promotion and evacuation on standard Young tableaux of rectangle and staircase shape,” Electron. J. Combin. 18 (2011), P18.

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    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The counterexample is valid. For k=6k=6, C=2C=2 satisfies Pon–Wang’s necessary condition C∣42C\mid 42 and C∤21C\nmid 21. Under the Edelman–Greene/Haiman correspondence, promotion corresponds to the cyclic/complement rotation ρ\rho (up to inverse, which does not affect fixed points of the square). If ρ2(a)=a\rho^2(a)=a, the reduced word must alternate between sas_a and s7−as_{7-a}, hence lies in a proper parabolic subgroup generated by at most two simple reflections. Such a word cannot represent w0∈S7w_0\in S_7, which requires all six simple generators. Therefore no tableau is fixed by ∂2\partial^2, so no promotion 2-cycle exists.

      This directly disproves sufficiency of Corollary 4.19’s condition for k>5k>5. I found no prior stronger/literature resolution in the checked sources.

      Novelty assessment

      TYPE1

      Classification rationale: The result appears genuinely new, but it is a very small counterexample: for k=6k=6, the admissible divisor C=2C=2 cannot occur. The proof is a short direct application of the Edelman–Greene/Haiman reduced-word model and in fact suggests an immediate broader “no 2-cycles” observation. This resolves Pon–Wang’s sufficiency question negatively, but it is too elementary and narrow to support a standalone journal paper.

      Literature check: I found no prior source stating this counterexample or a stronger classification of promotion cycle lengths for staircase SYT. Searches of the original Pon–Wang paper, related promotion/staircase/reduced-word literature, arXiv records, GitHub, and recent promotion-orbit papers found only the original open question and adjacent results: rectangular-tableau promotion, rowmotion/promotion, type-B reduced-word cyclic sieving, promotion digraphs, and orbit lengths for rectangular/near-hook cases. None addresses the SYT(sc6)SYT(sc_6) two-cycle obstruction or the sufficiency failure.

      Citation: Background/open question: Steven Pon and Qiang Wang, “Promotion and evacuation on standard Young tableaux of rectangle and staircase shape,” Electron. J. Combin. 18 (2011), P18. Also relevant: Edelman–Greene, “Balanced tableaux,” Adv. Math. 63 (1987), 42–99; Haiman, “Dual equivalence with applications,” Discrete Math. 99 (1992), 79–113.

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