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

Language | Haskell2010 |

Gelfand-Tsetlin patterns and Kostka numbers.

Gelfand-Tsetlin patterns (or tableaux) are triangular arrays like

[ 3 ] [ 3 , 2 ] [ 3 , 1 , 0 ] [ 2 , 0 , 0 , 0 ]

with both rows and columns non-increasing non-negative integers. Note: these are in bijection with the semi-standard Young tableaux.

If we add the further restriction that
the top diagonal reads `lambda`

,
and the diagonal sums are partial sums of `mu`

, where `lambda`

and `mu`

are two
partitions (in this case `lambda=[3,2]`

and `mu=[2,1,1,1]`

),
then the number of the resulting patterns
or tableaux is the Kostka number `K(lambda,mu)`

.
Actually `mu`

doesn't even need to the be non-increasing.

## Synopsis

- kostkaNumber :: Partition -> Partition -> Int
- kostkaNumberReferenceNaive :: Partition -> Partition -> Int
- kostkaNumbersWithGivenLambda :: forall coeff. Num coeff => Partition -> Map Partition coeff
- kostkaNumbersWithGivenMu :: Partition -> Map Partition Int
- type GT = [[Int]]
- asciiGT :: GT -> ASCII
- kostkaGelfandTsetlinPatterns :: Partition -> Partition -> [GT]
- kostkaGelfandTsetlinPatterns' :: Partition -> [Int] -> [GT]
- countKostkaGelfandTsetlinPatterns :: Partition -> Partition -> Int
- iteratedPieriRule :: Num coeff => [Int] -> Map Partition coeff
- iteratedPieriRule' :: Num coeff => Partition -> [Int] -> Map Partition coeff
- iteratedPieriRule'' :: Num coeff => (Partition, coeff) -> [Int] -> Map Partition coeff
- iteratedDualPieriRule :: Num coeff => [Int] -> Map Partition coeff
- iteratedDualPieriRule' :: Num coeff => Partition -> [Int] -> Map Partition coeff
- iteratedDualPieriRule'' :: Num coeff => (Partition, coeff) -> [Int] -> Map Partition coeff

# Kostka numbers

kostkaNumber :: Partition -> Partition -> Int Source #

Kostka numbers (via counting Gelfand-Tsetlin patterns). See for example http://en.wikipedia.org/wiki/Kostka_number

`K(lambda,mu)==0`

unless `lambda`

dominates `mu`

:

[ mu | mu <- partitions (weight lam) , kostkaNumber lam mu > 0 ] == dominatedPartitions lam

kostkaNumberReferenceNaive :: Partition -> Partition -> Int Source #

Very naive (and slow) implementation of Kostka numbers, for reference.

kostkaNumbersWithGivenLambda :: forall coeff. Num coeff => Partition -> Map Partition coeff Source #

Lists all (positive) Kostka numbers `K(lambda,mu)`

with the given `lambda`

:

kostkaNumbersWithGivenLambda lambda == Map.fromList [ (mu , kostkaNumber lambda mu) | mu <- dominatedPartitions lambda ]

It's much faster than computing the individual Kostka numbers, but not as fast as it could be.

kostkaNumbersWithGivenMu :: Partition -> Map Partition Int Source #

Lists all (positive) Kostka numbers `K(lambda,mu)`

with the given `mu`

:

kostkaNumbersWithGivenMu mu == Map.fromList [ (lambda , kostkaNumber lambda mu) | lambda <- dominatingPartitions mu ]

This function uses the iterated Pieri rule, and is relatively fast.

# Gelfand-Tsetlin patterns

kostkaGelfandTsetlinPatterns' :: Partition -> [Int] -> [GT] Source #

Generates all Kostka-Gelfand-Tsetlin tableau, that is, triangular arrays like

[ 3 ] [ 3 , 2 ] [ 3 , 1 , 0 ] [ 2 , 0 , 0 , 0 ]

with both rows and column non-increasing such that
the top diagonal read lambda (in this case `lambda=[3,2]`

) and the diagonal sums
are partial sums of mu (in this case `mu=[2,1,1,1]`

)

The number of such GT tableaux is the Kostka number K(lambda,mu).

countKostkaGelfandTsetlinPatterns :: Partition -> Partition -> Int Source #

This returns the corresponding Kostka number:

countKostkaGelfandTsetlinPatterns lambda mu == length (kostkaGelfandTsetlinPatterns lambda mu)

# The iterated Pieri rule

iteratedPieriRule :: Num coeff => [Int] -> Map Partition coeff Source #

Computes the Schur expansion of `h[n1]*h[n2]*h[n3]*...*h[nk]`

via iterating the Pieri rule.
Note: the coefficients are actually the Kostka numbers; the following is true:

Map.toList (iteratedPieriRule (fromPartition mu)) == [ (lam, kostkaNumber lam mu) | lam <- dominatingPartitions mu ]

This should be faster than individually computing all these Kostka numbers.

iteratedPieriRule' :: Num coeff => Partition -> [Int] -> Map Partition coeff Source #

Iterating the Pieri rule, we can compute the Schur expansion of
`h[lambda]*h[n1]*h[n2]*h[n3]*...*h[nk]`

iteratedDualPieriRule :: Num coeff => [Int] -> Map Partition coeff Source #

Computes the Schur expansion of `e[n1]*e[n2]*e[n3]*...*e[nk]`

via iterating the Pieri rule.
Note: the coefficients are actually the Kostka numbers; the following is true:

Map.toList (iteratedDualPieriRule (fromPartition mu)) == [ (dualPartition lam, kostkaNumber lam mu) | lam <- dominatingPartitions mu ]

This should be faster than individually computing all these Kostka numbers.
It is a tiny bit slower than `iteratedPieriRule`

.