Timeline for Decomposing the conjugacy representation of Sym$(n)$ for small $n$

6 events

when toggle format	what		by	license	comment
Jan 10, 2014 at 14:39	history	edited	Richard Stanley	CC BY-SA 3.0	added 324 characters in body
Jan 10, 2014 at 14:27	history	edited	Richard Stanley	CC BY-SA 3.0	added 324 characters in body
Jan 10, 2014 at 3:10	comment	added	Richard Stanley		This can be expressed in terms of plethysms of certain explicit symmetric functions, but there is no known combinatorial description. See Scharf and Thibon, A Hopf-algebra approach to inner plethysm, Remark 2.13. There is another decomposition of the character $\chi$ of the conjugacy action, namely, $\chi=\sum_{\lambda\vdash n}(\chi^\lambda)^2$ (where $\chi^\lambda$ is the irreducible character indexed by $\lambda$), but it is also not known how to decompose $(\chi^\lambda)^2$.
Jan 9, 2014 at 16:55	comment	added	Alexander Chervov		Is there some "combinatorial" description of natural subrepresenations - those which come from functions on conjugacy classes ? I mean: consider functions on each conjugacy class - they are subrepresentation - the classes are indexed by Young diagram. Irreps also. So from one Young diagram we should some list of "descent" diagrams ? Is there some description of this in terms of diagrams ?
Jan 9, 2014 at 2:59	vote	accept	Peter Dukes
Jan 9, 2014 at 2:12	history	answered	Richard Stanley	CC BY-SA 3.0