Generalized Foulkes' Conjecture and tableaux construction

symmetric group;wreath product

Vessenes, Rebecca Angel ; Wales, David B.

University:California Institute of Technology

Department:Physics, Mathematics and Astronomy

关键词: symmetric group;    wreath product;   

Others  :  https://thesis.library.caltech.edu/1870/1/AppendixA.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

Foulkes conjectured thatfor 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.