We make some progresses on Saxl Conjecture. Firstly, we show that the probability that a partition is comparable in dominance order to the staircase partition tends to zero as the staircase partition grows. Secondly, for partitions whose Durfee size is $k$ where $k\ge 3$, by semigroup property, we show that there exists a number $n_k$ such that if the tensor squares of the first $n_k$ staircase partitions contain all irreducible representations corresponding to partitions with Durfee size $k$, then all tensor squares contain partitions with Durfee size $k$. Especially, we show that $n_3=14$ and $n_4=28$. Furthermore, with the help of computer we show that the Saxl Conjecture is true for all triple hooks (i.e. partitions with Durfee size 3). Similar results for chopped square and caret shapes are also discussed.

我们在萨克斯猜想上取得了一些进展。首先，我们表明，随着楼梯间隔的增长，分区在主导顺序上与楼梯间隔相当的可能性趋于零。其次，对于Durfee大小为$ķ$ 在哪里 $ķ\ge 3$，通过semigroup属性，我们表明存在一个数字 ${ñ}_{ķ}$ 这样，如果第一个的张量平方 ${ñ}_{ķ}$ 楼梯分区包含与Durfee大小的分区对应的所有不可约表示 $ķ$，则所有张量平方都包含Durfee大小的分区 $ķ$。特别是，我们表明${ñ}_3=14$ 和 ${ñ}_4=28$。此外，借助计算机，我们证明了Saxl猜想对于所有三重钩子（即Durfee大小为3的分区）都是正确的。还讨论了切碎的正方形和插入符号形状的类似结果。