Saxl Conjecture for triple hooks
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by
Xin Li
2021
Abstract
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≥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. 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.
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