子词序同调中的反射表示
The reflection representation in the homology of subword order
AI总结：
该研究探讨对称群在子词序秩选择子偏序集上的同调表示，证明有界长度词同调模的张量幂分解，揭示反射表示同调的组合性质与对偶性，并给出Frobenius特征的相关结果及非负性猜想。
摘要：
我们研究对称群在子词序的秩选择子偏序集上的同调表示。我们证明，在大小为$n$的字母表上，有界长度词的同调模可分解为以分划$(n-1,1)$为指标的$S_n$-不可约表示$S_{(n-1,1)}$的张量幂之和，作为特例重新得到了Björner和Stanley关于长度至多为$k$的词的定理。对于任意秩，我们证明其同调是反射表示$S_{(n-1,1)}$的正张量幂的整数组合，并猜想该组合系数非负。我们发现了删除单个秩时同调中存在的一种奇特对偶性。我们证明，子词序的秩选择链上的作用是$S_{(n-1,1)}$的张量幂的非负整数组合，且其Frobenius特征是$h$-正的，支撑在集合$T_{1}(n)=\{h_λ: λ=(n-r, 1^r), r\ge 1\}$上。我们最具确定性的结果描述了：任意秩集的同调的Frobenius特征加上或减去一个Schur函数$s_{(n-1,1)}$，是集合$T_{2}(n)=\{h_λ: λ=(n-r, 1^r), r\ge 2\}$的整数组合。我们猜想该组合系数非负，并在特定情形下证明了这一结论。
英文摘要：
We investigate the homology representation of the symmetric group on rank-selected subposets of subword order. We show that the homology module for words of bounded length, over an alphabet of size $n,$ decomposes into a sum of tensor powers of the $S_n$-irreducible $S_{(n-1,1)}$ indexed by the partition $(n-1,1),$ recovering, as a special case, a theorem of Björner and Stanley for words of length at most $k.$ For arbitrary ranks we show that the homology is an integer combination of positive tensor powers of the reflection representation $S_{(n-1,1)}$, and conjecture that this combination is nonnegative. We uncover a curious duality in homology in the case when one rank is deleted. We prove that the action on the rank-selected chains of subword order is a nonnegative integer combination of tensor powers of $S_{(n-1,1)}$, and show that its Frobenius characteristic is $h$-positive and supported on the set $T_{1}(n)=\{h_λ: λ=(n-r, 1^r), r\ge 1\}.$ Our most definitive result describes the Frobenius characteristic of the homology for an arbitrary set of ranks, plus or minus one copy of the Schur function $s_{(n-1,1)},$ as an integer combination of the set $T_{2}(n)=\{h_λ: λ=(n-r, 1^r), r\ge 2\}.$ We conjecture that this combination is nonnegative, establishing this fact for particular cases.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
