Saxl Conjecture for triple hooks

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 $k\geq3$, 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 $k$. Specially, 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.
We are grateful to the editors and reviewers whose suggestions improve this paper greatly. To appear in Disc. Math