长度至多为四的划分的拉伸 Littlewood-Richardson 系数的正性
Positivity of stretched Littlewood-Richardson coefficients for partitions of length at most five
AI总结：
研究长度至多为四的划分的拉伸 Littlewood-Richardson 系数的正性，通过结构性证明，利用周期一扩张和局部方法，证明相关猜想，还得出两个秩均匀结果，并简化一般猜想，给出精确障碍及验证脚本。
AI中文摘要：
对于划分λ、μ、ν，Littlewood-Richardson 系数扩展为函数P(t)=c(tν;tλ,tμ)，根据 Derksen 和 Weyman 的定理，它是t的多项式。King、Tollu 和 Toumazet 猜想P没有负系数，该猜想仅在值至多为2时已知，其他情况未知。我们证明了对于所有部分数量至多为四的三元组的情况。证明是结构性的：周期一扩张将 Berline-Vergne 局部 Ehrhart 公式从整数扩张转移回有理蜂巢多面体，将陈述简化为验证由秩四菱形法线跨越的每个二维横向锥的 Berline-Vergne 权重的正性（最小权重为1/9）。未使用顶点完整性和顶点锥的幺模性，实际上在秩四时就已经不成立。相同的局部方法产生两个秩均匀的结果：对于任何秩的每个全维蜂巢多面体，前四个 Ehrhart 系数是正的，并且每个全维五部分蜂巢除了可能的线性系数外所有系数都是正的。最后，我们将一般猜想简化为一个特定于蜂巢的有效性陈述，并记录排除标准捷径的精确障碍。所有有限验证都伴有独立的重放脚本。
英文摘要：
For partitions lambda, mu, nu the Littlewood-Richardson coefficient stretches to a function P(t) = c(t*nu; t*lambda, t*mu) which, by a theorem of Derksen and Weyman, is a polynomial in t. King, Tollu and Toumazet conjectured that P has no negative coefficient. The conjecture is known only for c <= 2 and is otherwise open. We prove it for all triples whose parts number at most five. For at most four parts the proof is structural: a period-one dilation transfers the Berline-Vergne local Ehrhart formula from an integral dilate back to the rational hive polytope, and every two-dimensional transverse cone spanned by rank-four rhombus normals has positive weight (minimum 1/9). Five parts need two further ingredients. A closed-chamber transfer lemma, resting on Rassart's polynomiality of the coefficients on the closed cones of a chamber complex, reduces positivity for a fixed number of parts to full-dimensional hives. For full-dimensional five-part hives all coefficients but the linear one were already known to be positive; the linear one is where pointwise positivity of the local weights fails -- an edge cone of weight -347/109824 occurs on an actual hive -- and its nonnegativity is proved by an exact global certificate: a rational correction supported on 6,903 two-face types, balanced by the Minkowski closure of each two-face polygon, which makes every one of the 87,127 closed edge cones adjacent to them nonnegative. The same local method yields a rank-uniform consequence: for every full-dimensional hive polytope of any rank the top four Ehrhart coefficients are positive. We close by recording the exact general frontier and the obstructions that rule out the standard shortcuts. All finite verifications are exact and accompanied by independent replay scripts.
