Safe Haskell | None |
---|---|

Language | Haskell2010 |

Young tableaux and similar gadgets.

See e.g. William Fulton: Young Tableaux, with Applications to Representation theory and Geometry (CUP 1997).

The convention is that we use the English notation, and we store the tableaux as lists of the rows.

That is, the following standard Young tableau of shape [5,4,1]

1 3 4 6 7 2 5 8 10 9

is encoded conveniently as

[ [ 1 , 3 , 4 , 6 , 7 ] , [ 2 , 5 , 8 ,10 ] , [ 9 ] ]

- type Tableau a = [[a]]
- asciiTableau :: Show a => Tableau a -> ASCII
- _tableauShape :: Tableau a -> [Int]
- tableauShape :: Tableau a -> Partition
- tableauWeight :: Tableau a -> Int
- dualTableau :: Tableau a -> Tableau a
- tableauContent :: Tableau a -> [a]
- hooks :: Partition -> Tableau (Int, Int)
- hookLengths :: Partition -> Tableau Int
- rowWord :: Tableau a -> [a]
- rowWordToTableau :: Ord a => [a] -> Tableau a
- columnWord :: Tableau a -> [a]
- columnWordToTableau :: Ord a => [a] -> Tableau a
- isLatticeWord :: [Int] -> Bool
- isSemiStandardTableau :: Tableau Int -> Bool
- semiStandardYoungTableaux :: Int -> Partition -> [Tableau Int]
- countSemiStandardYoungTableaux :: Int -> Partition -> Integer
- isStandardTableau :: Tableau Int -> Bool
- standardYoungTableaux :: Partition -> [Tableau Int]
- countStandardYoungTableaux :: Partition -> Integer

# Basic stuff

asciiTableau :: Show a => Tableau a -> ASCII Source

ASCII diagram of a tableau

_tableauShape :: Tableau a -> [Int] Source

tableauShape :: Tableau a -> Partition Source

The shape of a tableau

tableauWeight :: Tableau a -> Int Source

Number of entries

dualTableau :: Tableau a -> Tableau a Source

The dual of the tableau is the mirror image to the main diagonal.

tableauContent :: Tableau a -> [a] Source

The content of a tableau is the list of its entries. The ordering is from the left to the right and then from the top to the bottom

hooks :: Partition -> Tableau (Int, Int) Source

An element `(i,j)`

of the resulting tableau (which has shape of the
given partition) means that the vertical part of the hook has length `i`

,
and the horizontal part `j`

. The *hook length* is thus `i+j-1`

.

Example:

> mapM_ print $ hooks $ toPartition [5,4,1] [(3,5),(2,4),(2,3),(2,2),(1,1)] [(2,4),(1,3),(1,2),(1,1)] [(1,1)]

hookLengths :: Partition -> Tableau Int Source

# Row and column words

rowWord :: Tableau a -> [a] Source

The *row word* of a tableau is the list of its entry read from the right to the left and then
from the top to the bottom.

rowWordToTableau :: Ord a => [a] -> Tableau a Source

*Semistandard* tableaux can be reconstructed from their row words

columnWord :: Tableau a -> [a] Source

The *column word* of a tableau is the list of its entry read from the bottom to the top and then from the left to the right

columnWordToTableau :: Ord a => [a] -> Tableau a Source

*Standard* tableaux can be reconstructed from either their column or row words

isLatticeWord :: [Int] -> Bool Source

Checks whether a sequence of positive integers is a *lattice word*,
which means that in every initial part of the sequence any number `i`

occurs at least as often as the number `i+1`

# Semistandard Young tableaux

isSemiStandardTableau :: Tableau Int -> Bool Source

A tableau is *semistandard* if its entries are weekly increasing horizontally
and strictly increasing vertically

semiStandardYoungTableaux :: Int -> Partition -> [Tableau Int] Source

Semistandard Young tableaux of given shape, "naive" algorithm

countSemiStandardYoungTableaux :: Int -> Partition -> Integer Source

Stanley's hook formula (cf. Fulton page 55)

# Standard Young tableaux

isStandardTableau :: Tableau Int -> Bool Source

A tableau is *standard* if it is semistandard and its content is exactly `[1..n]`

,
where `n`

is the weight.

standardYoungTableaux :: Partition -> [Tableau Int] Source

Standard Young tableaux of a given shape. Adapted from John Stembridge, http://www.math.lsa.umich.edu/~jrs/software/SFexamples/tableaux.

countStandardYoungTableaux :: Partition -> Integer Source

hook-length formula