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Total positivity for Grassmannians and amplituhedra

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total-positivity-for-grassmannians-and-amplituhedra-3Algebraic Geometrymath.AGmath.COposed by Steven Neil Karprecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

It is an interesting open problem to classify all fixed points of the twist map, and to determine whether Vk,nV_{k,n} is the only totally positive fixed point.

Context

Candidate 3 of the open problems stated in "Total positivity for Grassmannians and amplituhedra", 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: Interpreting the question in the standard Marsh–Scott sense: for a full-rank k×nk\times n matrix A=(v1,…,vn)A=(v_1,\dots,v_n) with cyclic kk-minors nonzero, the twist τ(A)=(w1,…,wn)\tau(A)=(w_1,\dots,w_n) is defined by

    wiTvi=1,wiTvi+1=⋯=wiTvi+k−1=0w_i^T v_i=1,\qquad w_i^T v_{i+1}=\cdots=w_i^T v_{i+k-1}=0

    (indices modulo nn). The question asks whether the cyclically symmetric point Vk,nV_{k,n} is the only totally positive fixed point of τ\tau. This is the testable uniqueness assertion in the quoted open problem.

    Result: The uniqueness assertion is false. There is a totally positive fixed point in Gr⁡>0(3,9)\operatorname{Gr}_{>0}(3,9) different from V3,9V_{3,9}.

    Let

    c=9−212,c=\frac{9-\sqrt{21}}2,

    so c2−9c+15=0c^2-9c+15=0 and 11/5<c<5/211/5<c<5/2. Define

    A=(1001339c−1724c−483c−3010−3−8c−93−3c16−9c−c0013c−38c−973c1).A=\begin{pmatrix} 1&0&0&1&3&3&9c-17&24c-48&3c-3\\ 0&1&0&-3&-8&c-9&3-3c&16-9c&-c\\ 0&0&1&3c-3&8c-9&7&3&c&1 \end{pmatrix}.

    A direct expansion of all ordered 3×33\times3 minors shows that their values lie in

    {1,3,c,7,8,18,3c−3,8c−9,9−c,9c−16,24c−48,9c−17,\{1,3,c,7,8,18,3c-3,8c-9,9-c,9c-16,24c-48,9c-17, 63c−135,165c−357,431c−936,63c−128,63c−127,1125c−2448,165c−336,21c−24}.63c-135,165c-357,431c-936,63c-128,63c-127, 1125c-2448,165c-336,21c-24\}.

    Since 11/5<c<5/211/5<c<5/2, every number listed is strictly positive. Thus X=rowspan⁡(A)∈Gr⁡>0(3,9)X=\operatorname{rowspan}(A)\in \operatorname{Gr}_{>0}(3,9).

    Now set

    B=(1003103c1).B=\begin{pmatrix}1&0&0\\3&1&0\\3&c&1\end{pmatrix}.

    Using c2=9c−15c^2=9c-15, direct multiplication gives, for the columns viv_i of AA,

    viTBvi=1,viTBvi+1=viTBvi+2=0v_i^T B v_i=1,\qquad v_i^T B v_{i+1}=v_i^T B v_{i+2}=0

    for all i∈Z/9Zi\in\mathbb Z/9\mathbb Z. Hence the ii-th twist column is BTviB^T v_i. Therefore

    τ(A)=BTA.\tau(A)=B^T A.

    Since B∈GL3(R)B\in GL_3(\mathbb R), rowspan⁡(τ(A))=rowspan⁡(A)\operatorname{rowspan}(\tau(A))=\operatorname{rowspan}(A), so XX is a fixed point of the twist.

    Finally, X≠V3,9X\neq V_{3,9}. The point V3,9V_{3,9} is cyclically symmetric, so after normalizing Δ123=1\Delta_{123}=1, one has Δ124=Δ235\Delta_{124}=\Delta_{235}. For the matrix above,

    Δ124=3c−3,Δ235=3,\Delta_{124}=3c-3,\qquad \Delta_{235}=3,

    and c≠2c\neq2. Thus XX is a distinct totally positive fixed point.

    Citation: No prior counterexample is used here. The twist definition is the standard one from Marsh–Scott, “Twists of Plücker coordinates as dimer partition functions,” Comm. Math. Phys. 341 (2016).

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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 KNOWN

      PASS

      The construction is a valid counterexample to the uniqueness part of the open problem. The matrix has full rank, the listed finite minor check establishes total positivity, and the identities viTBvi=1v_i^T Bv_i=1, viTBvi+1=viTBvi+2=0v_i^T Bv_{i+1}=v_i^T Bv_{i+2}=0 imply that the Marsh–Scott twist is BTAB^TA. Since BB is invertible, this gives the same Grassmannian point, so it is fixed by the twist. Finally Δ124=3c−3≠3=Δ235\Delta_{124}=3c-3\neq 3=\Delta_{235}, whereas V3,9V_{3,9} is cyclically symmetric, so the point is distinct from V3,9V_{3,9}. This does not classify all fixed points, but it rigorously disproves uniqueness.

      Novelty assessment

      KNOWN

      Classification rationale: The accepted counterexample is essentially a lift to Gr⁡>0(3,9)\operatorname{Gr}_{>0}(3,9) of a known non-regular projectively self-dual nonagon. For k=3k=3, twist-fixed points correspond to projective polygons whose vertex ii is dual to the side through vertices i+1,i+2i+1,i+2, i.e. to 33-self-dual 99-gons in the terminology of Fuchs–Tabachnikov. They prove that the moduli space of 33-self-dual 99-gons has dimension gcd⁡(3,9)−1=2\gcd(3,9)-1=2, and explicitly display non-regular examples. Convex real examples arbitrarily near the regular nonagon give totally positive Grassmannian points after positive rescaling of the vertex lifts.

      Literature check: I checked the twist/Grassmannian literature around Marsh–Scott, Karp, Muller–Speyer, Weng, and Shen–Weng, and also searched for the equivalent projective-geometry formulation. The decisive prior reference is Fuchs–Tabachnikov, which predates the Grassmannian twist formulation. Their mm-self-duality definition matches the k=3k=3 twist indexing: m=3m=3 maps a vertex to the side determined by the next two vertices. Their Theorem 1 / Proposition 13 gives a positive-dimensional family for (m,n)=(3,9)(m,n)=(3,9), and the paper notes “two 3-self-dual nonagons” in Figure 6.

      Citation: Dmitry Fuchs and Serge Tabachnikov, “Self-dual polygons and self-dual curves,” Functional Analysis and Other Mathematics 2 (2008), 203–220; arXiv:0707.1048.

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