Symmetrizing tableaux and the 5th case of the Foulkes conjecture. (May 2017)

• Record Type: Journal Article
• Authors: Cheung, Man-Wai; Ikenmeyer, Christian; Mkrtchyan, Sevak
• Is Part Of: Journal of symbolic computation. Volume 80:Part 3(2017)
• Page Start: 833
• Page End: 843
• Publication Date: 2017-05
• Subjects: 20C30 -- 20C40 -- 14-04; Representation theory of the symmetric group -- Plethysm -- Foulkes conjecture -- Foulkes–Howe conjecture -- Chow variety; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.09.002
• Languages: English
• ISSNs: 0747-7171

Abstract

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.