论文标题
子词顺序同源的反射表示
The reflection representation in the homology of subword order
论文作者
论文摘要
我们研究了对称组的同源性表示子词顺序的级别子题材。 我们表明，在大小$ n，$的字母上，有界长度的单词的同源模块 分解成$ s_n $ -rirreducibirabil $ s _ {（n-1,1）} $的张量幂的总和，分区$（n-1,1）索引，$恢复，特殊情况，特殊情况，是Björner和Stanley的一个定理，大多数$ k。表示$ s _ {（n-1,1）} $，并猜想这种组合是无负的。在删除一个等级的情况下，我们发现同源性的奇怪二元性。 我们证明，子词顺序排列的链条上的动作是$ s _ {（n-1,1）} $的张量功率的非负整数组合，并证明其frobenius特征为$ h $ postiver，并在集合$ t $ t_ {1}}（n）= \ ge / ge / ge / ge / n ge（n ge /geλ：λ=（r）上， 1 \}。$ 我们最明确的结果描述了一组任意等级的同源性的Frobenius特征，加上或减去Schur函数的一个副本$ s _ {（n-1,1）}，$作为集合的整数组合 $ t_ {2}（n）= \ {h_λ：λ=（n-r，1^r），r \ ge 2 \}。$我们猜想这种组合是非负的，在特定情况下确定了这一事实。
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.
