Generalized Foulkes' Conjecture and tableaux construction

Citation

Vessenes, Rebecca Angel (2004) Generalized Foulkes' Conjecture and tableaux construction. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-05192004-121256

Abstract

Foulkes conjectured that for n=ab and a <= b, every irreducible module occurring as a constituent in $1_{Sym{b}wrSym{a}}^{Sym{n}}$ occurs with greater or equal multiplicity in $1_{Sym{a}wrSym{b}}^{Sym{n}}$. We generalize part of this to say those irreducibles also occur in $1_{Sym{d}wrSym{c}}^{Sym{n}}$, where cd=n and c,d >= a. We prove the generalized conjecture for a=2 and a=3, by explicitly constructing the corresponding tableaux. We also prove the multiplicity constraint for certain cases. For these proofs we develop a theory of construction conditions for tableaux giving rise to $Sym{b}wrSym{a}$ modules and in doing so, completely classify all such tableaux for a=2 and a=3.

Thesis Files