For background, see linear representation theory of symmetric group:S3.
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Item |
Value
|
Degrees of irreducible representations over a splitting field (and in particular over ) |
1,1,2 maximum: 2, lcm: 2, number: 3 sum of squares: 6, quasirandom degree: 1
|
Schur index values of irreducible representations |
1,1,1
|
Smallest ring of realization for all irreducible representations (characteristic zero) |
|
Minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero) |
(hence, it is a rational representation group)
|
Condition for being a splitting field for this group |
Any field of characteristic not two or three is a splitting field. In particular, and are splitting fields.
|
Minimal splitting field in characteristic |
The prime field
|
Smallest size splitting field |
field:F5, i.e., the field of five elements.
|
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Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2 or 3, except in the last two columns, where we consider what happens in characteristic 2 and characteristic 3.
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Basic stuff
Up to equivalence, there are three irreducible representations of symmetric group:S3 in characteristic zero: the one-dimensional trivial representation, the one-dimensional sign representation (that sends every permutation to its sign), and the standard representation of symmetric group:S3, a two-dimensional representation.