Register to reply

Basic Symmetric Group Representation Question

by zer0skill
Tags: basic, representation, symmetric
Share this thread:
zer0skill
#1
Feb15-12, 03:09 PM
P: 1
If you consider the permutation representation of Sn in ℂ^n, i.e the transformation which takes a permutation into the operator which uses it to permute the coordinates of a vector, then of course the subspace such that every coordinate of the vector is the same is invariant under the representation. Also, the subspace in which all coordinates sum to zero is invariant. But are there any others that are independent of these ones?
Phys.Org News Partner Science news on Phys.org
Mysterious source of ozone-depleting chemical baffles NASA
Water leads to chemical that gunks up biofuels production
How lizards regenerate their tails: Researchers discover genetic 'recipe'
conquest
#2
Feb15-12, 04:31 PM
P: 117
The irreducible representations of the symmetric group in n letters are actually quite well understood and to find them all actually for a given vector space over C. They are even paramtrized by partitions of n. I couldn't give the answer explicitly very quickly, but just look for books on it keeping in mind the terms Specht module and Young diagram.
morphism
#3
Feb15-12, 06:29 PM
Sci Advisor
HW Helper
P: 2,020
No, there are no others: the two subspaces you mention are irreducible, have dimensions 1 and n-1 (resp.), and intersect in zero.


Register to reply

Related Discussions
Symmetric group question Calculus & Beyond Homework 0
A basic question on representation theory Advanced Physics Homework 0
Representation of finite group question Linear & Abstract Algebra 8
Basic matrix representation question Special & General Relativity 11
Generating Set for the Symmetric Group - Question Linear & Abstract Algebra 8