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combinat

 character
 compute character table for a symmetric group
 Chi
 compute Chi function for partitions of symmetric group

 Calling Sequence character(n) Chi(lambda, rho)

Parameters

 n - non-negative integer lambda - partition of n; non-decreasing list of positive integers rho - partition of n; non-decreasing list of positive integers

Description

 • 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 non-trivial 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 one-to-one correspondence between the partitions of n and the non-equivalent irreducible representations of G.  There is also a one-to-one 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

$\mathrm{Χ}\left({p}_{m-i+1},{p}_{j}\right)$

 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.

Examples

 > $\mathrm{with}\left(\mathrm{combinat}\right):$
 > $\mathrm{Chi}\left(\left[1,2\right],\left[1,1,1\right]\right)$
 ${2}$ (1)