Supercharacter theories for alternating group:A5

This article gives specific information, namely, supercharacter theories, about a particular group, namely: alternating group:A5.
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This page discusses the various possible supercharacter theories for alternating group:A5. Thus, it builds on a thorough understanding of the element structure of alternating group:A5, subgroup structure of alternating group:A5, and linear representation theory of alternating group:A5.

We describe the group $A_{5}$ as the alternating group on $\{1,2,3,4,5\}$, and elements of the group are described by means of their cycle decompositions.

Character table

Below, the character table of $A_{5}$ is given. This table is crucial for understanding the possible supercharacter theories.

Representation/conjugacy class representative and size	$()$ (size 1)	$(1,2)(3,4)$ (size 15)	$(1,2,3)$ (size 20)	$(1,2,3,4,5)$ (size 12)	$(1,2,3,5,4)$ (size 12)
trivial	1	1	1	1	1
restriction of standard	4	0	1	-1	-1
irreducible five-dimensional	5	1	-1	0	0
one irreducible constituent of restriction of exterior square of standard	3	-1	0	$({\sqrt {5}}+1)/2$	$(-{\sqrt {5}}+1)/2$
other irreducible constituent of restriction of exterior square of standard	3	-1	0	$(-{\sqrt {5}}+1)/2$	$({\sqrt {5}}+1)/2$

Supercharacter theories

Note that for each of the supercharacter tables presented, the supercharacter is the smallest positive integer linear combination of the characters in the block that takes constant values on each superconjugacy class.

Summary

Quick description of supercharacter theory	Number of blocks of conjugacy classes = number of blocks of irreducible representations	Block sizes for conjugacy classses (in number of conjugacy class terms) (should add up to 5, the total number of conjugacy classes)	Block sizes for conjugacy classes (in number of elements terms) (should add up to 60, the order of the group)	Block sizes for irreducible representations (in number of representations terms) (should add up to 5, the total number of conjugacy classes)	Block sizes for irreducible representations (in sum of squares of degrees terms) (should add up to 60, the order of the group)
ordinary character theory	5	1,1,1,1,1	1,12,12,15,20	1,1,1,1,1	1,9,9,16,25
all non-identity elements form one block	2	1,4	1,59	1,4	1,59
supercharacter theory for the automorphism group action, or equivalently, for the Galois group action	4	1,1,1,2	1,15,20,24	1,1,1,2	1,16,18,25