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Linear representation theory of symmetric group:S3

311 bytes removed, 01:32, 18 July 2011
Summary information
===Summary information===
Below is summary information on irreducible representationsthat are absolutely irreducible, i.e. Note that a particular representation may make sense, and be they remain irreduciblein any bigger field, only for certain kinds of fields -- see the "Values not allowed for field characteristic" and "Criterion for in particular are irreducible in a [[splitting field" columns to see ]]. We assume that the condition characteristic of the field must satisfy for is not 2 or 3, except in the representation to be irreducible therelast two columns, where we consider what happens in characteristic 2 and characteristic 3.
{| class="sortable" border="1"
! Name of representation type !! Number of representations of this type !! Values not allowed for field characteristic !! Criterion for field !! What happens over a splitting field? !! Kernel !! [[Degree of a linear representation|Degree]] !! [[Schur index]] !! What happens Criterion for field !! Kernel !! Quotient by reducing the <math>\mathbb{Z}</math>-kernel (on which it descends to a faithful representation over bad characteristics) !! Characteristic 2?!! Characteristic 3
|-
| trivial || 1 || -- 1 || any 1 || remains the same any || whole group || 1 [[trivial group]] || 1 works || --works
|-
| sign || 1 || -- 1 || any 1 || remains the same any || [[A3 in S3]] (unless the characteristic is two, in which case it is the whole group) || 1 || 1 [[cyclic group:Z2]] || there are no ''bad characteristics''works, but it is noteworthy that in characteristic two, this becomes same as trivial representation|| works
|-
| [[standard representation of symmetric group:S3|standard (two-dimensional irreducible)]] || 1 || 3 2 || any 1 || remains the same any || trivial subgroup, i.e., it is faithful || 2 || 1 || When the <math>\mathbb{Z}</math>-representation is mapped to [[fieldsymmetric group:F3S3]], we get a representation that is || works || indecomposable but not irreducible.
|}
 
===Trivial representation===
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