YoungTableaux

Documentation for YoungTableaux.

YoungTableaux.YoungTableaux — Module

YoungTableaux

See the demo here

Young Tableaux can be constructed as follows:

julia> using YoungTableaux

julia> YoungTableau([[1, 3, 4], [2, 5], [6]])
3×3 YoungTableau{Int64}
┌───┬───┬───┐
│ 1 │ 3 │ 4 │
├───┼───┼───┘
│ 2 │ 5 │
├───┼───┘
│ 6 │
└───┘

Alternatively they can also be constructed from a permutation:

julia> π = [4, 6, 3, 8, 1, 2, 7, 5]
8-element Vector{Int64}:
 4
 6
 3
 8
 1
 2
 7
 5

julia> P, Q = rs_pair(π)
(YoungTableau{Int64}([[1, 2, 5], [3, 6, 7], [4, 8]]), YoungTableau{Int64}([[1, 2, 4], [3, 6, 7], [5, 8]]))

julia> P
3×3 YoungTableau{Int64}
┌───┬───┬───┐
│ 1 │ 2 │ 5 │
├───┼───┼───┤
│ 3 │ 6 │ 7 │
├───┼───┼───┘
│ 4 │ 8 │
└───┴───┘

julia> Q
3×3 YoungTableau{Int64}
┌───┬───┬───┐
│ 1 │ 2 │ 4 │
├───┼───┼───┤
│ 3 │ 6 │ 7 │
├───┼───┼───┘
│ 5 │ 8 │
└───┴───┘

Broadcasting works just like with arrays!

julia> P .+ Q
3×3 YoungTableau{Int64}
┌────┬────┬────┐
│  2 │  4 │  9 │
├────┼────┼────┤
│  6 │ 12 │ 14 │
├────┼────┼────┘
│  9 │ 16 │
└────┴────┘

Partitions are just like Young Tableaux, but without entries. When iterating a partition only true is returned. Indices of the square can be computed using eachindex.

julia> Partition([3, 3, 2])
3×3 Partition
┌───┬───┬───┐
│   │   │   │
├───┼───┼───┤
│   │   │   │
├───┼───┼───┘
│   │   │
└───┴───┘

julia> eachindex(Partition([3, 3, 2]))
3×3 YoungTableaux.EachIndexOf{Partition}
┌─────┬─────┬─────┐
│ 1,1 │ 1,2 │ 1,3 │
├─────┼─────┼─────┤
│ 2,1 │ 2,2 │ 2,3 │
├─────┼─────┼─────┘
│ 3,1 │ 3,2 │
└─────┴─────┘

The partition corresponding to a Young Tableaux can be requested using shape, which will return a lazy wrapper:

julia> shape(P)
3×3 YoungTableaux.PartitionOf{YoungTableau{Int64}}
┌───┬───┬───┐
│   │   │   │
├───┼───┼───┤
│   │   │   │
├───┼───┼───┘
│   │   │
└───┴───┘

Addition and substraction work as defined in Macdonald: "Symmetric Functions and Hall Polynomials"

julia> shape(P) - Partition([2, 1])
3×3 SkewPartition
┌───┬───┬───┐
│   │   │ * │
├───┼───┼───┤
│   │ * │ * │
├───┼───┼───┘
│ * │ * │
└───┴───┘

julia> shape(P) + Partition([2, 1])
3×5 Partition
┌───┬───┬───┬───┬───┐
│   │   │   │   │   │
├───┼───┼───┼───┼───┘
│   │   │   │   │
├───┼───┼───┴───┘
│   │   │
└───┴───┘

There is also initial support for conjugating diagrams using the adjoint operator like with regular matrices. It is currently implemented lazily but this may be subject to change.

julia> P'
3×3 YoungTableaux.ConjugateDiagram{Int64, YoungTableau{Int64}}
┌───┬───┬───┐
│ 1 │ 3 │ 4 │
├───┼───┼───┤
│ 2 │ 6 │ 8 │
├───┼───┼───┘
│ 5 │ 7 │
└───┴───┘
source
YoungTableaux.AbstractShape — Type
abstract type AbstractShape <: AbstractDiagram{Bool} end

A diagram only describing a certain shape, potentially with holes such as a skew partition

source
YoungTableaux.rows — Function
rows(::AbstractDiagram)

Returns an array containing all rows of a diagram. Must be implemented by any subtype of AbstractDiagram. Unfilled diagrams should return a Bool

source