Saxl Conjecture for triple hooks release_5s7s2avnunde5dx2vei5h3l3ka

by Xin Li

Released as a article .

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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Date   2021-02-18
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arXiv  1811.10967v4
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