# Supercharacter theories for symmetric group:S4

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This page discusses the various possible supercharacter theories for symmetric group:S4. Thus, it builds on a thorough understanding of the element structure of symmetric group:S4, subgroup structure of symmetric group:S4, and linear representation theory of symmetric group:S4.

We describe the group $S_4$ as the symmetric group on $\{ 1,2,3,4 \}$, and elements of the group are described by means of their cycle decompositions.

## Character table

Below, the character table of $S_4$ is given. This table is crucial for understanding the possible supercharacter theories.

Representation/Conjugacy class representative and size $()$ (identity element) (size 1) $(1,2)(3,4)$ (size 3) $(1,2)$ (size 6) $(1,2,3,4)$ (size 6) $(1,2,3)$ (size 8)
Trivial representation 1 1 1 1 1
Sign representation 1 1 -1 -1 1
Irreducible representation of degree two with kernel of order four 2 2 0 0 -1
Standard representation 3 -1 1 -1 0
Product of standard and sign representations 3 -1 -1 1 0

## 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

There is a total of five possible supercharacter theories:

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 24, 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 24, the order of the group)
ordinary character theory 1 5 1,1,1,1,1 1,3,6,6,8 1,1,1,1,1 1,1,4,9,9
all non-identity elements form one block 1 2 1,4 1,23 1,4 1,23
corresponds to normal series with normal V4 in S4 as the middle subgroup 1 3 1,1,3 1,3,20 1,2,2 1,5,18
corresponds to normal series with A4 in S4 as the middle subgroup 1 3 1,2,2 1,11,12 1,1,3 1,1,22
corresponds to full chief series (has both normal V4 in S4 and A4 in S4) 1 4 1,1,1,2 1,3,8,12 1,1,1,2 1,1,4,18
Total (5 rows) 5 (equals total number of supercharacter theories)

### Ordinary character theory

One extreme case of a supercharacter theory is the one where we simply use blocks of size one both on the conjugacy class side and the linear representation side. The "supercharacter table" in this case is just the usual character table:

Representation/Conjugacy class representative and size $()$ (identity element) (size 1) $(1,2)(3,4)$ (size 3) $(1,2)$ (size 6) $(1,2,3,4)$ (size 6) $(1,2,3)$ (size 8)
Trivial representation 1 1 1 1 1
Sign representation 1 1 -1 -1 1
Irreducible representation of degree two with kernel of order four 2 2 0 0 -1
Standard representation 3 -1 1 -1 0
Product of standard and sign representations 3 -1 -1 1 0

### Supercharacter theory with all non-identity elements in one block

This is the other extreme supercharacter theory: all the non-identity elements form a single block, and all the nontrivial representations form a single block. The supercharacter table is as follows:

Supercharacter/superconjugacy class and size $()$ (identity element) (size 1, 1 conjugacy class) all non-identity elements (size 23, 4 conjugacy classes)
character of trivial representation 1 1
regular representation minus trivial representation = sum of all nontrivial characters weighted by degree = sign + 2(two-dimensional irreducible) + 3(standard) + 3(product of standard and sign) 23 -1

### Supercharacter theory with a block for the Klein four-subgroup as kernel

Further information: supercharacter theory corresponding to a normal series

This supercharacter theory corresponds to the normal series:

trivial subgroup $\le$ $\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$ (normal V4 in S4) $\le$ whole group

The superconjugacy classes are the set differences between adjacent members of the normal series, and the blocks on the representation side are the collections of representations whose kernel contains one member but not the preceding member of the normal series.

Explicitly, the superconjugacy classes are as follows:

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
non-identity elements of normal V4 in S4 1 $(1,2)(3,4)$ 3 3
all elements outside normal V4 in S4 3 $(1,2)$, $(1,2,3,4)$, $(1,2,3)$ 6,6,8 20
Total (3 superconjugacy classes) 5 (equals number of conjugacy classes in the whole group) -- 1,3,6,6,8 (equals conjugacy class size statistics of the whole group) 24 (equals order of the whole group)

The blocks for representations, and corresponding supercharacters, are as follows:

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
trivial character 1 character of trivial representation (multiplicity 1) 1 1 1
character of regular representation of quotient by normal V4 in S4 composed with quotient map minus trivial character 2 character of sign representation (multiplicity 1), character of two-dimensional irreducible representation of $S_4$ (mutliplicity 2) 1,2 1,4 5
(1/3) character of (regular representation of whole group - regular representation of quotient by Klein four-subgroup (composed with quotient map)) 2 character of standard representation (multiplicity 1), character of product of standard and sign representation (multiplicity 1) 3,3 9,9 18
Total 5 (equals number of irreducible representations equals number of conjugacy classes) -- 1,1,2,3,3 (degrees of irreducible representations) 1,1,4,9,9 (squares of degrees of irreducible representations) 24 (equals order of the whole group)

The supercharacter table is:

Supercharacter/superconjugacy class representative and size $()$ (identity element) (size 1, 1 conjugacy class) $(1,2)(3,4)$ (size 3, 1 conjugacy class) all other elements (size 20, 3 conjugacy classes)
character of trivial representation 1 1 1
regular representation of quotient by Klein four-subgroup (composed with quotient map) - trivial representation = (sign representation) + 2 (two-dimensional irreducible representation of $S_4$) 5 5 -1
(1/3) (regular representation of whole group - regular representation of quotient by Klein four-subgroup (composed with quotient map)) = (standard) + (product of standard and sign) 6 -2 0

Note that the multiplication by $1/3$ in the last row is to maintain our convention of choosing the smallest possible positive combination of the characters that works.

### Supercharacter theory with a block for the alternating group as kernel

Further information: supercharacter theory corresponding to a normal series

This supercharacter theory corresponds to the normal series:

trivial subgroup $\le$ A4 in S4 $\le$ whole group

The superconjugacy classes are the set differences between adjacent members of the normal series, and the blocks on the representation side are the collections of representations whose kernel contains one member but not the preceding member of the normal series.

Explicitly, the superconjugacy classes are:

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
non-identity even permutations, i.e., non-identity elements of A4 in S4 2 $(1,2)(3,4)$, $(1,2,3)$ 3,8 11
odd permutations, i.e., elements outside A4 in S4 2 $(1,2)$, $(1,2,3,4)$ 6,6 12
Total (3 superconjugacy classes) 5 (equals number of conjugacy classes in the whole group) -- 1,3,6,6,8 (equals conjugacy class size statistics of the whole group) 24 (equals order of the whole group)

The blocks for representations, and corresponding supercharacters, are as follows:

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
trivial character 1 character of trivial representation (multiplicity 1) 1 1 1
character of regular representation of quotient by A4 in S4 composed with quotient map minus trivial character 1 character of sign representation (multiplicity 1) 1 1 1
character of (regular representation of whole group - regular representation of quotient by A4 in S4 (composed with quotient map)) 3 character of two-dimensional irreducible (multiplicity 2), character of standard representation (multiplicity 3), character of product of standard and sign representation (multiplicity 3) 2,3,3 4,9,9 22
Total 5 (equals number of irreducible representations equals number of conjugacy classes) -- 1,1,2,3,3 (degrees of irreducible representations) 1,1,4,9,9 (squares of degrees of irreducible representations) 24 (equals order of the whole group)

The supercharacter table is:

Supercharacter/superconjugacy class $()$ (identity element) (size 1, 1 conjugacy class) non-identity elements of A4 in S4, i.e., non-identity even permutations (size 11, 2 conjugacy classes, representatives $(1,2)(3,4)$ and $(1,2,3)$ all other elements, i.e., odd permutations (size 12, 2 conjugacy classes, representatives $(1,2)$ and $(1,2,3,4)$)
character of trivial representation 1 1 1
regular representation of quotient by A4 in S4 (composed with quotient map) - trivial representation = sign representation 1 1 -1
(regular representation of whole group - regular representation of quotient by A4 in S4 = 2(two-dimensional irreducible) + 3(standard) + 3(product of standard and sign) 22 -2 0

### Supercharacter theory with a block for the full chief series

Further information: supercharacter theory corresponding to a normal series

This supercharacter theory corresponds to the normal series:

trivial subgroup $\le$ $\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$ (normal V4 in S4) $\le$ A4 in S4 $\le$ whole group

The superconjugacy classes are the set differences between adjacent members of the normal series, and the blocks on the representation side are the collections of representations whose kernel contains one member but not the preceding member of the normal series.

Explicitly, the superconjugacy classes are:

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
non-identity elements of normal V4 in S4 1 $(1,2)(3,4)$ 3 3
elements of A4 in S4 outside normal V4 in S4 1 $(1,2,3)$ 8 8
odd permutations, i.e., elements outside A4 in S4 2 $(1,2)$, $(1,2,3,4)$ 6,6 12
Total (4 superconjugacy classes) 5 (equals number of conjugacy classes in the whole group) -- 1,3,6,6,8 (equals conjugacy class size statistics of the whole group) 24 (equals order of the whole group)

The blocks for representations, and corresponding supercharacters, are as follows:

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
trivial character 1 character of trivial representation (multiplicity 1) 1 1 1
character of regular representation of quotient by A4 in S4 composed with quotient map minus trivial character 1 character of sign representation (multiplicity 1) 1 1 1
(1/2) character of (regular representation of quotient by normal V4 in S4 (composed with quotient map) - regular representation of quotient by A4 in S4 (composed with quotient map)) 1 character of two-dimensional irreducible (multiplicity 1) 2 4 4
(1/3) character of (regular representation of whole group - regular representation of quotient by Klein four-subgroup (composed with quotient map)) 2 character of standard representation (multiplicity 1), character of product of standard and sign representation (multiplicity 1) 3,3 9,9 18
Total (4 superconjugacy classes) 5 (equals number of irreducible representations equals number of conjugacy classes) -- 1,1,2,3,3 (degrees of irreducible representations) 1,1,4,9,9 (squares of degrees of irreducible representations) 24 (equals order of the whole group)

The supercharacter table is:

Supercharacter/superconjugacy class $()$ (identity element) (size 1, 1 conjugacy class) class of $(1,2)(3,4)$ (1 class, 3 elements) class of $(1,2,3)$ (1 class, 8 elements) all other elements, i.e., odd permutations (size 12, 2 conjugacy classes, representatives $(1,2)$ and $(1,2,3,4)$)
character of trivial representation 1 1 1 1
regular representation of quotient by A4 in S4 (composed with quotient map) - trivial representation = sign representation 1 1 1 -1
(1/2) (regular representation of quotient by Klein four-subgroup (composed with quotient map) - regular representation of quotient by A4 in S4 (composed with quotient map)) = (two-dimensional irreducible) 2 2 -1 0
(1/3)(regular representation of whole group - regular representation of quotient by Klein four-subgroup (composed with quotient map)) = (standard) + (product of standard and sign) 6 -2 0 0