Symmetrizing tableaux and the 5th case of the Foulkes conjecture. (May 2017) - Cheung, Man-Wai; Ikenmeyer, Christian; Mkrtchyan, Sevak
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.