alternating representation



For S nS_n a symmetric group, its alternating representation is the 1-dimensional linear representation

altS nGL(1) alt \;\; S_n \longrightarrow GL(1)

which sends even permutations to +1+1 and odd permutations to 1-1.

More generally, for GG any finite group and HGH \subset G a subgroup of index 2, then the corresponding alternating representation of GG sends elements of HH to +1+1 and all other elements to -1.

This reduces to the previous special case by setting GS nG \coloneqq S_n and HA nS nH \coloneqq A_n \subset S_n the alternating group.


For instance