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Differential posets and restriction in critical groups

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differential-posets-and-restriction-in-critical-groupsNumber Theorymath.COmath.NTposed by Ayush Agarwal, Christian Gaetzrecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

Working in the differential tower of groups AıSA \imath\mathfrak{S} with A abelian of order r, when k ≤n the critical group K(V(UkDk)n)=K(IndAıSn−kAıSn1)K(V(U^{k}D^{k})_{n})=K(Ind_{A \imath\mathfrak{S}_{n-k}}^{A \imath\mathfrak{S}_{n}}\mathbb{1}) is given, as a list of elementary divisors, by:(1pn−k,(rkn!(n−k)!)pn−2pn−k+pn−2k,rkn!(n−k)!ei),\left(1^{p_{n-k}},\left(r^{k}\frac{n!}{(n-k)!}\right)^{p_{n}-2p_{n-k}+p_{n-2k}},r^{k}\frac{n!}{(n-k)!}e_{i}\right),where the exponents involving rank sizes denote multiplicities, and where eie_{i} ranges over the non-unit elementary divisors in the critical group K(V(DkUk)n−k)K(V(D^{k}U^{k})_{n-k}) .

Context

Candidate 1 of the open problems stated in "Differential posets and restriction in critical groups", 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: Conjecture 5.7 asserts that for A≀SA\wr \mathfrak S, AA abelian of order rr, k≤nk\le n, with pj=∣(Yr)j∣p_j=|(Y^r)_j|, the critical group of

    V(UkDk)n=Ind⁡A≀Sn−kA≀Sn1V(U^kD^k)_n=\operatorname{Ind}_{A\wr \mathfrak S_{n-k}}^{A\wr \mathfrak S_n}\mathbf 1

    has elementary divisors

    (1pn−k,(rkn!(n−k)!)pn−2pn−k+pn−2k,rkn!(n−k)!ei),\left(1^{p_{n-k}}, \left(r^k\frac{n!}{(n-k)!}\right)^{p_n-2p_{n-k}+p_{n-2k}}, r^k\frac{n!}{(n-k)!}e_i\right),

    where eie_i are the non-unit elementary divisors of K(V(DkUk)n−k)K(V(D^kU^k)_{n-k}).

    Result: The conjecture is false as stated.

    Take r=1r=1, so AA is trivial and A≀Sn=SnA\wr\mathfrak S_n=\mathfrak S_n. Let n=4n=4, k=2k=2. Then p4=5p_4=5, p2=2p_2=2, p0=1p_0=1, and

    c=rkn!(n−k)!=12.c=r^k\frac{n!}{(n-k)!}=12.

    Using the rank-4 partition order

    (4),(3,1),(2,2),(2,1,1),(1,1,1,1),(4),(3,1),(2,2),(2,1,1),(1,1,1,1),

    one computes

    U2D2=(1211025341132311435201121).U^2D^2= \begin{pmatrix} 1&2&1&1&0\\ 2&5&3&4&1\\ 1&3&2&3&1\\ 1&4&3&5&2\\ 0&1&1&2&1 \end{pmatrix}.

    Thus the critical matrix is

    12I−U2D2=(11−2−1−10−27−3−4−1−1−310−3−1−1−4−37−20−1−1−211),12I-U^2D^2= \begin{pmatrix} 11&-2&-1&-1&0\\ -2&7&-3&-4&-1\\ -1&-3&10&-3&-1\\ -1&-4&-3&7&-2\\ 0&-1&-1&-2&11 \end{pmatrix},

    whose Smith normal form is

    diag⁡(1,1,12,60,0).\operatorname{diag}(1,1,12,60,0).

    So

    K(V(U2D2)4)≅Z/12Z⊕Z/60Z.K(V(U^2D^2)_4)\cong \mathbb Z/12\mathbb Z\oplus \mathbb Z/60\mathbb Z.

    For the lower group, on rank 22,

    D2U2=(7557),12I−D2U2=(5−5−55),D^2U^2= \begin{pmatrix} 7&5\\ 5&7 \end{pmatrix}, \qquad 12I-D^2U^2= \begin{pmatrix} 5&-5\\ -5&5 \end{pmatrix},

    with Smith normal form diag⁡(5,0)\operatorname{diag}(5,0). Hence the only non-unit eie_i is 55.

    The conjectured list is therefore

    (1p2,12p4−2p2+p0,12⋅5)=(1,1,12,12,60),(1^{p_2},12^{p_4-2p_2+p_0},12\cdot 5) =(1,1,12,12,60),

    which predicts an extra elementary divisor 1212. The actual list is

    (1,1,12,60).(1,1,12,60).

    Thus Conjecture 5.7 is disproved.

    The failure comes from the boundary term pn−2k=p0p_{n-2k}=p_0: here it contributes one extra 1212, although the lower critical group has no corresponding unit factor to account for it. A natural repair would need to replace pn−2kp_{n-2k} by the actual number of unit elementary divisors in K(V(DkUk)n−k)K(V(D^kU^k)_{n-k}), or impose additional hypotheses excluding this boundary behavior.

    Citation: Counterexample computed directly above; source statement is Agarwal–Gaetz, “Differential posets and restriction in critical groups,” Conjecture 5.7.

    Reviews

    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 proposed counterexample attacks the stated conjecture exactly. For r=1,n=4,k=2r=1,n=4,k=2, the computed U2D2U^2D^2, D2U2D^2U^2, and Smith normal forms are consistent. The actual critical group has finite elementary divisors (1,1,12,60)(1,1,12,60), while the conjectured formula predicts (1,1,12,12,60)(1,1,12,12,60). Thus it rigorously disproves the conjecture as stated.

      Novelty assessment

      TYPE1

      Classification rationale: This is a genuine small counterexample to Agarwal–Gaetz Conjecture 5.7, but it is a direct finite computation in the smallest nontrivial boundary case (r,n,k)=(1,4,2)(r,n,k)=(1,4,2). It does not give a repaired theorem or broader structural explanation. On its own it is best suited as an erratum/comment or as part of a larger correction, not a standalone combinatorics paper.

      Literature check: I found no existing public reference to this counterexample or to a stronger result resolving Conjecture 5.7. Searches covered the original arXiv/published record, Gaetz’s arXiv author page for follow-up work, related critical-group/differential-poset papers, and public web/code/forum traces using phrases such as “Conjecture 5.7” “Differential posets”, “Agarwal Gaetz critical groups”, “V(U^kD^k)”, “U^2D^2 critical group”, and the specific elementary divisors 12,6012,60. The visible literature still points back to the original conjectural statement, with no erratum or later correction located.

      Citation: Ayush Agarwal and Christian Gaetz, “Differential posets and restriction in critical groups,” Algebraic Combinatorics 2 (2019), no. 6, 1311–1327; arXiv:1710.08253.

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