三足蜘蛛的Schur正性
Schur positivity of three-legged spiders
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中文总结 AI 辅助
本文完整分类了三足蜘蛛的Schur正性，给出充要条件并确定唯一负系数，推广了已有结果，证明结合Pieri规则与归纳法。
中文摘要 AI 辅助
我们给出了三足蜘蛛Schur正性的完整分类。每条至少具有两条偶腿的蜘蛛都是Schur正的，而每条具有三条奇腿的蜘蛛都有一个负的Schur系数。若$a$为偶数且$b,c$为奇数，则蜘蛛$S(a,b,c)$是Schur正的当且仅当$a\le 5b+5c+2$。每当Schur正性不成立时，恰好存在一个负的Schur系数，我们明确确定了其值。这些结果推广了Thibon和Wang关于族$S(a,2,1)$和$S(a,4,1)$的结果，以及Wang和Wang关于$S(a,b,2)$的结果。证明结合了团蜘蛛的已知Schur正性结果、Pieri规则和同步归纳法。
英文摘要
We give a complete classification of Schur positivity for three-legged spiders. Every spider with at least two even legs is Schur positive, whereas every spider with three odd legs has a negative Schur coefficient. If $a$ is even and $b,c$ are odd, then the spider $S(a,b,c)$ is Schur positive if and only if $a\le 5b+5c+2$. Whenever Schur positivity fails, there is exactly one negative Schur coefficient, whose value we determine explicitly. These results extend those of Thibon and Wang for the families $S(a,2,1)$ and $S(a,4,1)$, and those of Wang and Wang for $S(a,b,2)$. The proof combines known Schur-positivity results for clique-spiders with Pieri's rule and simultaneous induction.
发表机构
- Beijing Institute of Technology（北京理工大学）
机构由 AI 辅助整理，请以论文原文为准。
