Linear representation theory of general affine group:GA(2,3)

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This article gives specific information, namely, linear representation theory, about a particular group, namely: general affine group:GA(2,3).
View linear representation theory of particular groups | View other specific information about general affine group:GA(2,3)

Summary

Item Value
degrees of irreducible representations over a splitting field (such as \mathbb{C} or \overline{\mathbb{Q}}) 1,1,2,2,2,3,3,4,8,8,16
grouped form: 1 (2 times), 2 (3 times), 3 (2 times), 4 (1 time),8 (2 times), 16 (1 time)
number: 11, maximum: 16, lcm: 48, sum of squares: 432

Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

Interpretation as general affine group of degree two

Compare and contrast with linear representation theory of general affine group of degree two over a finite field

GAP implementation

The degrees of irreducible representations can be calculated using GAP's CharacterDegrees and GA functions (the GA function is not in-built):

gap> CharacterDegrees(GA(2,3));
[ [ 1, 2 ], [ 2, 3 ], [ 3, 2 ], [ 4, 1 ], [ 8, 2 ], [ 16, 1 ] ]