The trivial or principal representation is a one-dimensional representation sending every element of the symmetric group to the identity matrix of order one. This representation makes sense over all fields, and its character is 1 on all elements
The sign representation is a one-dimensional representation sending every permutation to its ''sign'': the even permutations get sent to 1 and the odd permutations get sent to -1. The kernel of this representation (i.e. the permutations that get sent to one) is the alternating group: the unique cyclic subgroup of order three comprising permutations <math>(1,2,3)</math>, <math>(1,3,2)</math> and the identity permutation. The three permutations of order two all get sent to -1.
This representation makes sense over any field, but when the characteristic of the field is two, it is the same as the trivial representation.