Math.Combinat.Tableaux

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 ]
]

Basic stuff

type Tableau a = [[a]]

A tableau is simply represented as a list of lists.

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

ASCII diagram of a tableau

_tableauShape :: Tableau a -> [Int]

tableauShape :: Tableau a -> Partition

The shape of a tableau

tableauWeight :: Tableau a -> Int

Number of entries

dualTableau :: Tableau a -> Tableau a

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

tableauContent :: Tableau a -> [a]

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)

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

Row and column words

rowWord :: Tableau a -> [a]

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

Semistandard tableaux can be reconstructed from their row words

columnWord :: Tableau a -> [a]

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

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

isLatticeWord :: [Int] -> Bool

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

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

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

Semistandard Young tableaux of given shape, "naive" algorithm

countSemiStandardYoungTableaux :: Int -> Partition -> Integer

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

Standard Young tableaux

isStandardTableau :: Tableau Int -> Bool

A tableau is standard if it is semistandard and its content is exactly [1..n], where n is the weight.

standardYoungTableaux :: Partition -> [Tableau Int]

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

countStandardYoungTableaux :: Partition -> Integer

hook-length formula