combinat
character
compute character table for a symmetric group
Chi
compute Chi function for partitions of symmetric group
Calling Sequence
Parameters
Description
Examples
character(n)
Chi(lambda, rho)
n

nonnegative integer
lambda
partition of n; nondecreasing list of positive integers
rho
Given a group (G,*), a group of matrices (H,&*) homomorphic to G is termed a representation of G. A representation is said to be reducible if there exists a similarity transformation
$S:H\to XH{X}^{1}$
that maps all elements of H to the same nontrivial block diagonal structure. If a representation is not reducible, it is termed an irreducible representation.
Given two elements of the same conjugacy class in G, the traces of their corresponding matrices in any representation are equal. The character function Chi is defined such that Chi of a conjugacy class of an irreducible representation of a group is the trace of any matrix corresponding to a member of that conjugacy class.
Taking G to be the symmetric group on n elements, ${S}_{n}$, there is a onetoone correspondence between the partitions of n and the nonequivalent irreducible representations of G. There is also a onetoone correspondence between the partitions of n and the conjugacy classes of G.
The Maple function Chi works on symmetric groups. Chi(lambda, rho) will compute and return the trace of the matrices in the conjugacy class corresponding to the partition rho in the irreducible representation corresponding to the partition lambda, where lambda and rho are of type partition. Clearly, both rho and lambda must be partitions of the same number.
The function character(n) computes Chi(lambda, rho) for all partitions lambda and rho of n. Thus, it computes the character of all conjugacy classes for all irreducible representations of the symmetric group on n elements.
For partitions ${p}_{i}$ of n, in ascending lexicographical ordering, for example $\left[1\,1\,\mathrm{...}\,1\right],\mathrm{...},\left[n\right]$, the $i$,$j$th entry of the character table for ${S}_{n}$ is given by
$\mathrm{Chi}\left({p}_{mi+1}\,{p}_{j}\right)$
thus the row ordering is reversed. This is the standard layout as given in the book The Theory of Group Characters by D. E. Littlewood.
$\mathrm{with}\left(\mathrm{combinat}\right)\:$
$\mathrm{Chi}\left(\left[1\,2\right]\,\left[1\,1\,1\right]\right)$
${2}$
See Also
combinat[partition]
GroupTheory[CharacterTable]
type/partition
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