Provides some functions for working with Young diagrams.

[jiahao]

• partitionsequence: computes partition sequence for an integer partition
• a \ b creates the skew diagram for partitions (tuples) a, b
• isrimhook: checks if skew diagram is a rim hook
• leglength: computes leg length of rim hook
• character: computes character of irrep of Sn

Begins to address #4

[daviddelaat]

This is great @jiahao !

I had some ugly code for pulling the character tables from GAP lying around, so I compared the results. There seem to be small differences, however. (probably an error on my side though). Do you know what this is?

using JSON
require("youngdiagrams.jl")

immutable SymmetricGroup
    n::Int
end

gap(command::String) = readall(`echo $command` | `gap -q`)

function charactertable(g::SymmetricGroup)
    n = g.n
    t = gap("Display(Irr(CharacterTable(\"Sym($n)\")));")
    t = filter(x -> contains("-0123456789,[]", x), t)[6:end] # on some system the [6:end] should not be there...
    l = JSON.parse(t)
    m = Array(Int, length(l), length(l))
    for i = 1:length(l)
        for j = 1:length(l)
            m[i, j] = l[i][j]
        end
    end
    m
end

# Lists the partitions of the number n, the order is consistent with GAP
function partitions(n)
    if n == 1
        return Vector{Int}[[1]]
    end

    list = Vector{Int}[]

    for p in partitions(n-1)
        push!(list, [p, 1])
        if length(p) == 1 || p[end] < p[end-1]
            push!(list, [p[1:end-1], p[end]+1])
        end
    end        

    list
end

n = 5

println(charactertable(SymmetricGroup(n)))

pn = length(partitions(n))
m = zeros(Int,pn,pn)
for i = 1:pn
    for j = 1:pn
        m[i,j] = character(partitions(n)[i],partitions(n)[j])
    end
end
println(m)

The output:

Gap character table for S_5:
7x7 Int64 Array:
 1  -1   1   1  -1  -1   1
 4  -2   0   1   1   0  -1
 5  -1   1  -1  -1   1   0
 6   0  -2   0   0   0   1
 5   1   1  -1   1  -1   0
 4   2   0   1  -1   0  -1
 1   1   1   1   1   1   1
Young diagram character table for S_5:
7x7 Int64 Array:
 1  -1  1   1  -1  -1   1
 8  -2  0   1   1   0  -1
 4  -1  1  -1  -1   1   0
 4   0  0   0   0   0   1
 4   1  1  -1   1  -1   0
 2   2  0   1  -1   0  -1
 1   1  1   1   1   1   1

[jiahao]

@daviddelaat thanks for checking the code! I must confess to not being an expert and only testing the few examples given in the reference. This probably needs more debugging and I could use some help ;)

[daviddelaat]

I'm also not an expert on this (that's why I pull the tables from GAP instead of writing my own algorithm ;-) ), but I'll try to take a look at this over the weekend.

The smallest value where I think it might go wrong is character([3,1],[1,1,1,1]) which I think should give 3 but which gives 2.

[andrioni]

Thanks! I spent the last week away, but I'm back now. Great use of Unicode characters!

[andrioni]

Just a slight observation: I ended up renaming the types to use CamelCase in e533608175b6452afcb5ff24d960ae1ff93d0e9b for consistency.