# Changes

## Linear representation theory of symmetric group:S3

, 00:27, 30 July 2011
Orthogonality relations and numerical checks
==Orthogonality relations and numerical checks==

Recall that the degrees of irreducible representations are 1,1,2.
{| class="sortable" border="1"
! General statement !! Verification in this case
|-
| [[number of irreducible representations equals number of conjugacy classes]] || Both numbers are equal to 3.<br>As symmetric group [itex]S_n, n = 3[/itex]: both numbers are equal to the [[number of unordered integer partitions]] of 3.<br>As [itex]GL(2,q)[/itex], [itex]q = 2[/itex]: both numbers are equal to [itex]q^2 - 1 = 2^2 - 1 = 3[/itex].<br>As [itex]GA(1,q), q = 3[/itex]: Both numbers are equal to [itex]q = 3[/itex].
|-
| [[sufficiently large implies splitting]]: if the field has characteristic not dividing the order of the group and has primitive [itex]d^{th}[/itex] roots of unity for [itex]d[/itex] the exponent of the group, it is a splitting field. || In fact, for this group, ''any'' field of characteristic not 2 or 3 is a splitting field.