The sign representation of the symmetric group 
, for a positive
integer 
,
is the one-dimensional complex group representation
on 
given by
 |
where 
,
,
and 
is the signature of the permutation 
. Thus an even
permutation acts as 
and an odd permutation
acts as 
.
This is an irreducible representation;
its group character is 
(Etingof et al. 2011).
