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===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 there~~last 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 ~~[[~~field~~symmetric group:~~F3~~S3]]~~, we get a representation that is ~~|| works || indecomposable but not irreducible~~.~~

|}

===Trivial representation===

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