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Linear representation theory of symmetric group:S3

384 bytes added, 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.
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! 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 <math>S_n, n = 3</math>: both numbers are equal to the [[number of unordered integer partitions]] of 3.<br>As <math>GL(2,q)</math>, <math>q = 2</math>: both numbers are equal to <math>q^2 - 1 = 2^2 - 1 = 3</math>.<br>As <math>GA(1,q), q = 3</math>: Both numbers are equal to <math>q = 3</math>.
| [[sufficiently large implies splitting]]: if the field has characteristic not dividing the order of the group and has primitive <math>d^{th}</math> roots of unity for <math>d</math> 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.
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