# Supercharacter theories for 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 such supercharacter theories under automorphism group action 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 1 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 1 2 1,4 1,59 1,4 1,59
supercharacter theory for the automorphism group action, or equivalently, for the Galois group action 1 4 1,1,1,2 1,15,20,24 1,1,1,2 1,16,18,25
Total (3 rows) 3 (equals total number of supercharacter theories)

### Ordinary character theory

Here, the blocks for conjugacy classes all have size one and the blocks for irreducible representations all have size one. The supercharacter table is the same as the character table.

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 $(-\sqrt{5} + 1)/2$ $(\sqrt{5} + 1)/2$

### All non-identity elements form one block

There are two blocks of conjugacy classes: the identity element is one block, and all non-identity elements form the other block. There are two blocks of irreducible characters: the trivial character is one block, and all other characters form the other block. The supercharacter table is as follows:

Supercharacter/superconjugacy class and size $()$ (identity element) (size 1, 1 conjugacy class) all non-identity elements (size 59, 4 conjugacy classes)
character of trivial representation 1 1
regular representation minus trivial representation = sum of all nontrivial characters weighted by degree = 3 (three-dimensional) + 3 (other three-dimensional) + 4 (four-dimensional (standard)) + 5 (five-dimensional) 59 -1

### Supercharacter theory for automorphism group action

In this case, the two conjugacy classes of 5-cycles get fused into one superconjugacy class. All other conjugacy classes remain intact. On the character side, the two characters of degree three form one block. All other characters form separate blocks.

Below is an explicit description of the superconjugacy classes:

Description of superconjugacy class Number of conjugacy classes in it Representatives of conjugacy classes in it Sizes of conjugacy classes Total number of group elements
identity element $()$ 1 $()$ 1 1
double transposition 1 $(1,2)(3,4)$ 15 15
3-cycle 1 $(1,2,3)$ 20 20
5-cycle 2 $(1,2,3,4,5)$ and $(1,2,3,5,4)$ 12,12 24
Total (4 rows) 5 (equals number of conjugacy classes in the whole group) -- 1,12,12,15,20 (equals conjugacy class size statistics of the whole group) 60 (equals order of the whole group)

Below is an explicit description of the supercharacters and the blocks of characters:

Description of supercharacter Number of irreducible characters in it List of irreducible characters in it, with multiplicities used in supercharacter Degrees Squares of degrees Sum of squares of degrees
character of trivial representation 1 trivial (multiplicity 1) 1 1 1
restriction of standard 1 restriction of standard (multiplicity 1) 4 16 16
five-dimensional irreducible 1 five-dimensional irreducible (multiplicity 1) 5 25 25
full restriction of exterior square of standard 2 both three-dimensional irreducibles, each with multiplicity 1 3,3 18 18
Total (4 rows) 5 (equals number of conjugacy classes in the whole group) -- 1,3,3,4,5 (equals degrees of irreducible representations of the whole group) 1,9,9,16,25 60 (equals order of the whole group)

Below is the supercharacter table:

Supercharacter and superconjugacy class $()$ (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) (total size 24)
trivial 1 1 1 1
restriction of standard 4 0 1 -1
irreducible five-dimensional 5 1 -1 0
restriction of exterior square of standard = sum of the three-dimensional irreducibles 6 -2 0 1