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El grupo de pilas de arena de una multigráfica / The Sandpile group of a multigraph

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el-grupo-de-pilas-de-arena-de-una-multigrafica-the-sandpile-group-of-aNumber Theorymath.GRmath.NTposed by Carlos Alejandro Alfaro Montúfarrecorded: open · 1 machine check, unexamined

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Statement

This results lead us to establish a conjecture that gives us an algebraic and a combinatorial description of the sandpile group of the cone of the hypercube QdQ_d of dimension dd. More precisely, K(c(Qd))≅⨁i=1dZ2i+1(di)=Z3d⊕Z5(d2)⊕⋯⊕Z2d−1(dd−1)⊕Z2d+1.K(c(Q_d)) \cong \bigoplus_{i=1}^{d} \mathbb{Z}_{2i+1}^{\binom{d}{i}} = \mathbb{Z}_3^d \oplus \mathbb{Z}_5^{\binom{d}{2}} \oplus \cdots \oplus \mathbb{Z}_{2d-1}^{\binom{d}{d-1}} \oplus \mathbb{Z}_{2d+1}. and SP(c(Qd),s)=⨁{β∈{0,1}d}K~β≅⨁{β∈{0,1}d}Z2∣β∣+1=⨁i=1dZ2i+1(di),SP(c(Q_d), s) = \bigoplus_{\{\beta \in \{0,1\}^d\}} \tilde{K}_\beta \cong \bigoplus_{\{\beta \in \{0,1\}^d\}} \mathbb{Z}_{2|\beta|+1} = \bigoplus_{i=1}^{d} \mathbb{Z}_{2i+1}^{\binom{d}{i}}, where K~β={gβ(r,t)+(d−∣β∣)1∣0≤r,t≤d and r=∣β∣ or t=∣β∣}≅Z2∣β∣+1\tilde{K}_\beta = \{g_\beta(r, t) + (d - |\beta|)1 \mid 0 \le r, t \le d \text{ and } r = |\beta| \text{ or } t = |\beta|\} \cong \mathbb{Z}_{2|\beta|+1} for all β∈{0,1}d\beta \in \{0,1\}^d and $g_\beta(r, t)_a = \begin{cases} r & \text{if } \beta \cdot \mathbf{a} \text{ is even,} \ t & \text{if } \beta \cdot \mathbf{a} \text{ is odd.} \end{cases}$$

Context

Candidate 1 of the open problems stated in "El grupo de pilas de arena de una multigráfica / The Sandpile group of a multigraph", 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 statement: for d≥0d\ge 0, let QdQ_d be the hypercube on V={0,1}dV=\{0,1\}^d, and let c(Qd)c(Q_d) be its cone, with cone vertex ss used as sink. Then the sandpile group

    K(c(Qd),s)=ZV/ΔdZV,Δd=(d+1)I−A(Qd)=L(Qd)+I,K(c(Q_d),s)=\mathbb Z^{V}/\Delta_d\mathbb Z^{V}, \qquad \Delta_d=(d+1)I-A(Q_d)=L(Q_d)+I,

    satisfies

    K(c(Qd),s)≅⨁β∈{0,1}dZ/(2∣β∣+1)≅⨁i=1d(Z/(2i+1))(di).K(c(Q_d),s)\cong \bigoplus_{\beta\in\{0,1\}^d}\mathbb Z/(2|\beta|+1) \cong \bigoplus_{i=1}^{d}\left(\mathbb Z/(2i+1)\right)^{\binom di}.

    The β=0\beta=0 summand is trivial. The displayed “combinatorial” sets must be interpreted as sandpile classes, or equivalently as recurrent stabilizations of the listed representatives; as raw stable configurations the formula is not literal, since e.g. for d=2d=2, β=(1,0)\beta=(1,0), r=1,t=2r=1,t=2, a coordinate equals 3>d3>d.

    Result: The intended statement is true.

    For β∈{0,1}d\beta\in\{0,1\}^d, define

    χβ(a)=(−1)β⋅a,a∈{0,1}d,\chi_\beta(a)=(-1)^{\beta\cdot a},\qquad a\in\{0,1\}^d,

    where the dot product is mod 22. If AA is the adjacency matrix of QdQ_d, then

    (Aχβ)(a)=∑j=1dχβ(a+ej)=(d−2∣β∣)χβ(a).(A\chi_\beta)(a)=\sum_{j=1}^d \chi_\beta(a+e_j) =(d-2|\beta|)\chi_\beta(a).

    Hence

    Δdχβ=(L(Qd)+I)χβ=(2∣β∣+1)χβ.\Delta_d\chi_\beta=(L(Q_d)+I)\chi_\beta=(2|\beta|+1)\chi_\beta.

    Let H=(χβ(a))a,βH=(\chi_\beta(a))_{a,\beta} be the Walsh-Hadamard matrix. Then

    HTH=2dI.H^T H=2^d I.

    For every odd prime pp, H∈GL2d(Z(p))H\in GL_{2^d}(\mathbb Z_{(p)}). Therefore over Z(p)\mathbb Z_{(p)},

    ΔdH=Hdiag⁡(2∣β∣+1),\Delta_d H=H\operatorname{diag}(2|\beta|+1),

    so

    coker⁡Z(p)(Δd)≅⨁β∈{0,1}dZ(p)/(2∣β∣+1).\operatorname{coker}_{\mathbb Z_{(p)}}(\Delta_d) \cong \bigoplus_{\beta\in\{0,1\}^d} \mathbb Z_{(p)}/(2|\beta|+1).

    Also

    det⁡Δd=∏β(2∣β∣+1)\det\Delta_d=\prod_{\beta}(2|\beta|+1)

    is odd, so the sandpile group has odd order. Thus its pp-primary component is exactly the above localization. Comparing all odd primes gives

    K(c(Qd),s)≅⨁β∈{0,1}dZ/(2∣β∣+1),K(c(Q_d),s)\cong \bigoplus_{\beta\in\{0,1\}^d}\mathbb Z/(2|\beta|+1),

    and regrouping by ∣β∣=i|\beta|=i yields the conjectured algebraic decomposition.

    For the explicit generators, let

    uβ(a)={1,β⋅a even,0,β⋅a odd.u_\beta(a)= \begin{cases} 1,&\beta\cdot a\text{ even},\\ 0,&\beta\cdot a\text{ odd}. \end{cases}

    Since Δd1=1\Delta_d\mathbf 1=\mathbf 1, the constant vector is zero in the sandpile group, and

    2[uβ]=[χβ].2[u_\beta]=[\chi_\beta].

    Multiplication by 22 is an automorphism because the group has odd order, so the cyclic subgroup generated by [uβ][u_\beta] has order 2∣β∣+12|\beta|+1, and these subgroups form an internal direct sum.

    Finally,

    gβ(r,t)=t1+(r−t)uβ,g_\beta(r,t)=t\mathbf 1+(r-t)u_\beta,

    so in the sandpile group

    [gβ(r,t)+(d−∣β∣)1]=(r−t)[uβ].[g_\beta(r,t)+(d-|\beta|)\mathbf 1]=(r-t)[u_\beta].

    With k=∣β∣k=|\beta|, the allowed conditions r=kr=k or t=kt=k realize all residues

    −k,−k+1,…,k(mod2k+1).-k,-k+1,\dots,k \pmod{2k+1}.

    Hence the displayed K~β\widetilde K_\beta is precisely the cyclic subgroup generated by [uβ][u_\beta], of order 2∣β∣+12|\beta|+1. Thus

    SP(c(Qd),s)=⨁β∈{0,1}dK~β≅⨁i=1d(Z/(2i+1))(di),SP(c(Q_d),s)=\bigoplus_{\beta\in\{0,1\}^d}\widetilde K_\beta \cong \bigoplus_{i=1}^{d}\left(\mathbb Z/(2i+1)\right)^{\binom di},

    under the standard identification of sandpile classes with their unique recurrent representatives.

    Citation: No prior complete resolution is used here. Context source: C. A. Alfaro and C. E. Valencia, “On the Sandpile group of the cone of a graph,” Linear Algebra Appl. 436 (2012), 1154–1176.

    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 KNOWN

      PASS

      The proof correctly attacks the cone-of-hypercube sandpile group: the reduced Laplacian is Δ=(d+1)I−A=L(Qd)+I\Delta=(d+1)I-A=L(Q_d)+I, and the Walsh–Hadamard character basis gives eigenvalues 2∣β∣+12|\beta|+1. Since the Hadamard matrix is invertible over every odd localization and det⁡Δ\det\Delta is odd, the local cokernels determine the finite abelian group, yielding

      K(c(Qd))≅⨁βZ/(2∣β∣+1).K(c(Q_d))\cong \bigoplus_{\beta}\mathbb Z/(2|\beta|+1).

      The generator description is also justified at the level of sandpile classes: constants vanish, 2[uβ]=[χβ]2[u_\beta]=[\chi_\beta], and the listed gβ(r,t)g_\beta(r,t) realize all residues mod 2∣β∣+12|\beta|+1.

      The caveat about interpreting K~β\widetilde K_\beta as classes/recurrent stabilizations rather than literal raw stable configurations is necessary and reasonable. I found only related prior work on generators and on sandpile groups of hypercubes/Cayley graphs, not a prior complete resolution of this cone statement.

      Novelty assessment

      KNOWN

      Classification rationale: The accepted resolution is not new: the same cone-of-hypercube sandpile group decomposition is already proved in the source thesis itself, despite being phrased conjecturally in the abstract. The cited Theorem 2.56 and Corollary 2.57 give the combinatorial direct-sum description and the resulting algebraic isomorphism

      K(c(Qd))≅⨁i=1dZ2i+1(di).K(c(Q_d))\cong \bigoplus_{i=1}^d \mathbb Z_{2i+1}^{\binom di}.

      Literature check: I checked the arXiv/LAA paper by Alfaro–Valencia, indexed mirrors/search results for “sandpile group of the cone of the hypercube,” and the thesis metadata/context. The LAA paper mainly advertises generators for the cone of the hypercube, but the thesis named in the prompt contains the full statement as a proved main result, not an open conjecture. Thus the prover’s proof is a rediscovery.

      Citation: Carlos Alejandro Alfaro Montúfar, El grupo de pilas de arena de una multigráfica / The Sandpile group of a multigraph, Theorem 2.56 and Corollary 2.57. Context: C. A. Alfaro and C. E. Valencia, “On the Sandpile group of the cone of a graph,” Linear Algebra Appl. 436 (2012), 1154–1176.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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