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arXiv · 1509.03944

Symmetrizing Tableaux and the 5th case of the Foulkes Conjecture

Man-Wai Cheung·Christian Ikenmeyer·Sevak Mkrtchyan
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Abstract

The Foulkes conjecture states that the multiplicities in the plethysm Sym^a(Sym^b V) are at most as large as the multiplicities in the plethysm Sym^b(Sym^a V) for all a <= b. This conjecture has been known to be true for a <= 4. The main result of this paper is its verification for a = 5. This is achieved by performing a combinatorial calculation on a computer and using a propagation theorem of Tom McKay from 2008. Moreover, we obtain a complete representation theoretic decomposition of the vanishing ideal of the 5th Chow variety in degree 5, we show that there are no degree 5 equations for the 6th Chow variety, and we also find some representation theoretic degree 6 equations for the 6th Chow variety.

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Man-Wai Cheung, Christian Ikenmeyer, Sevak Mkrtchyan. 2015-09-14. Symmetrizing Tableaux and the 5th case of the Foulkes Conjecture. https://arxiv.org/abs/1509.03944

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