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dual -- dual character



Returns the dual of a character, i.e., the character of the dual or contragredient representation.

The first argument is the original character.

Assuming the polynomial ring over which the character is defined has a coefficient field F which is a subfield of the complex numbers, then the second argument is the restriction of complex conjugation to F.

As an example, we construct a character of the alternating group $A_4$ considered as a subgroup of the symmetric group $S_4$. The conjugacy classes are represented by the identity, and the permutations $(12)(34)$, $(123)$, and $(132)$, in cycle notation. The character is constructed over the field $\mathbb{Q}[w]$, where $w$ is a primitive third root of unity. Complex conjugation restricts to $\mathbb{Q}[w]$ by sending $w$ to $w^2$. The character is concentrated in homological degree 1, and internal degree 2.

i1 : F = toField(QQ[w]/ideal(1+w+w^2))

o1 = F

o1 : PolynomialRing
i2 : R = F[x_1..x_4]

o2 = R

o2 : PolynomialRing
i3 : conj = map(F,F,{w^2})

o3 = map (F, F, {- w - 1})

o3 : RingMap F <-- F
i4 : X = character(R,4,hashTable {(1,{2}) => matrix{{1,1,w,w^2}}})

o4 = Character over R
     (1, {2}) => | 1 1 w -w-1 |

o4 : Character
i5 : X' = dual(X,conj)

o5 = Character over R
     (-1, {-2}) => | 1 1 -w-1 w |

o5 : Character

If working over coefficient fields of positive characteristic or if one wishes to avoid defining conjugation, one may replace the second argument by a list containing a permutation $\pi$ of the integers $1,\dots,r$, where $r$ is the number of conjugacy classes of the group. The permutation $\pi$ is defined as follows: if $g$ is an element of the $j$-th conjugacy class, then $g^{-1}$ is an element of the $\pi (j)$-th class.

In the case of $A_4$, the identity and $(12)(34)$ are their own inverses, while $(123)^{-1} = (132)$. Therefore the permutation $\pi$ is the transposition exchanging 3 and 4. Hence the dual of the character in the example above may also be constructed as follows, with $\pi$ represented in one-line notation by a list passed as the second argument.

i6 : perm = {1,2,4,3}

o6 = {1, 2, 4, 3}

o6 : List
i7 : dual(X,perm) === X'

o7 = true

The page characterTable contains some motivation for using conjugation or permutations of conjugacy classes when dealing with characters.

See also

Ways to use dual :

For the programmer

The object dual is a method function with options.