-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/
-- | Jack, zonal, Schur and skew Schur polynomials
--
-- This library can compute Jack polynomials, zonal polynomials, Schur
-- and skew Schur polynomials. It also provides some utilities for
-- symmetric polynomials.
@package jackpolynomials
@version 1.4.2.0
-- | Evaluation of Jack polynomials, zonal polynomials, Schur polynomials
-- and skew Schur polynomials. See README for examples and references.
module Math.Algebra.Jack
type Partition = [Int]
-- | Evaluation of Jack polynomial
jack' :: [Rational] -> Partition -> Rational -> Char -> Rational
-- | Evaluation of zonal polynomial
zonal' :: [Rational] -> Partition -> Rational
-- | Evaluation of Schur polynomial
schur' :: [Rational] -> Partition -> Rational
-- | Evaluation of a skew Schur polynomial
skewSchur' :: [Rational] -> Partition -> Partition -> Rational
-- | Evaluation of Jack polynomial
jack :: forall a. (Eq a, C a) => [a] -> Partition -> a -> Char -> a
-- | Evaluation of zonal polynomial
zonal :: (Eq a, C a) => [a] -> Partition -> a
-- | Evaluation of Schur polynomial
schur :: forall a. C a => [a] -> Partition -> a
-- | Evaluation of a skew Schur polynomial
skewSchur :: forall a. (Eq a, C a) => [a] -> Partition -> Partition -> a
module Math.Algebra.Jack.HypergeoPQ
-- | Inefficient hypergeometric function of a matrix argument (for testing
-- purpose)
hypergeoPQ :: (Eq a, C a) => Int -> [a] -> [a] -> [a] -> a
-- | Computation of symbolic Jack polynomials, zonal polynomials, Schur
-- polynomials and skew Schur polynomials. See README for examples and
-- references.
module Math.Algebra.JackPol
-- | Symbolic Jack polynomial
jackPol' :: Int -> Partition -> Rational -> Char -> Spray Rational
-- | Symbolic zonal polynomial
zonalPol' :: Int -> Partition -> Spray Rational
-- | Symbolic Schur polynomial
schurPol' :: Int -> Partition -> Spray Rational
-- | Symbolic skew Schur polynomial
skewSchurPol' :: Int -> Partition -> Partition -> Spray Rational
-- | Symbolic Jack polynomial
jackPol :: forall a. (Eq a, C a) => Int -> Partition -> a -> Char -> Spray a
-- | Symbolic zonal polynomial
zonalPol :: forall a. (Eq a, C a) => Int -> Partition -> Spray a
-- | Symbolic Schur polynomial
schurPol :: forall a. (Eq a, C a) => Int -> Partition -> Spray a
-- | Symbolic skew Schur polynomial
skewSchurPol :: forall a. (Eq a, C a) => Int -> Partition -> Partition -> Spray a
-- | Computation of Jack polynomials with a symbolic Jack parameter. See
-- README for examples and references.
module Math.Algebra.JackSymbolicPol
-- | Jack polynomial with symbolic Jack parameter
jackSymbolicPol :: forall a. (Eq a, C a) => Int -> Partition -> Char -> ParametricSpray a
-- | Jack polynomial with symbolic Jack parameter
jackSymbolicPol' :: Int -> Partition -> Char -> ParametricQSpray
-- | A Jack polynomial can have a very long expression in the canonical
-- basis. A considerably shorter expression is obtained by writing the
-- polynomial as a linear combination of the monomial symmetric
-- polynomials instead, which is always possible since Jack polynomials
-- are symmetric. This is the initial motivation of this module. But now
-- it contains more stuff dealing with symmetric polynomials.
module Math.Algebra.SymmetricPolynomials
-- | Checks whether a spray defines a symmetric polynomial.
--
--
-- >>> -- note that the sum of two symmetric polynomials is not symmetric
--
-- >>> -- if they have different numbers of variables:
--
-- >>> spray = schurPol' 4 [2, 2] ^+^ schurPol' 3 [2, 1]
--
-- >>> isSymmetricSpray spray
--
isSymmetricSpray :: (C a, Eq a) => Spray a -> Bool
-- | Monomial symmetric polynomial
--
--
-- >>> putStrLn $ prettySpray' (msPolynomial 3 [2, 1])
-- (1) x1^2.x2 + (1) x1^2.x3 + (1) x1.x2^2 + (1) x1.x3^2 + (1) x2^2.x3 + (1) x2.x3^2
--
msPolynomial :: (C a, Eq a) => Int -> Partition -> Spray a
-- | Power sum polynomial
--
--
-- >>> putStrLn $ prettyQSpray (psPolynomial 3 [2, 1])
-- x^3 + x^2.y + x^2.z + x.y^2 + x.z^2 + y^3 + y^2.z + y.z^2 + z^3
--
psPolynomial :: (C a, Eq a) => Int -> Partition -> Spray a
-- | Complete symmetric homogeneous polynomial
--
--
-- >>> putStrLn $ prettyQSpray (cshPolynomial 3 [2, 1])
-- x^3 + 2*x^2.y + 2*x^2.z + 2*x.y^2 + 3*x.y.z + 2*x.z^2 + y^3 + 2*y^2.z + 2*y.z^2 + z^3
--
cshPolynomial :: (C a, Eq a) => Int -> Partition -> Spray a
-- | Elementary symmetric polynomial.
--
--
-- >>> putStrLn $ prettyQSpray (esPolynomial 3 [2, 1])
-- x^2.y + x^2.z + x.y^2 + 3*x.y.z + x.z^2 + y^2.z + y.z^2
--
esPolynomial :: (C a, Eq a) => Int -> Partition -> Spray a
-- | Symmetric polynomial as a linear combination of monomial symmetric
-- polynomials.
msCombination :: C a => Spray a -> Map Partition a
-- | Symmetric polynomial as a linear combination of power sum polynomials.
-- Symmetry is not checked.
psCombination :: forall a. (Eq a, C a) => Spray a -> Map Partition a
-- | Symmetric polynomial as a linear combination of power sum polynomials.
-- Same as psCombination but with other constraints on the base
-- ring of the spray.
psCombination' :: forall a. (Eq a, C Rational a, C a) => Spray a -> Map Partition a
-- | Symmetric polynomial as a linear combination of complete symmetric
-- homogeneous polynomials. Symmetry is not checked.
cshCombination :: forall a. (Eq a, C a) => Spray a -> Map Partition a
-- | Symmetric polynomial as a linear combination of complete symmetric
-- homogeneous polynomials. Same as cshCombination but with
-- other constraints on the base ring of the spray.
cshCombination' :: forall a. (Eq a, C Rational a, C a) => Spray a -> Map Partition a
-- | Symmetric polynomial as a linear combination of elementary symmetric
-- polynomials. Symmetry is not checked.
esCombination :: forall a. (Eq a, C a) => Spray a -> Map Partition a
-- | Symmetric polynomial as a linear combination of elementary symmetric
-- polynomials. Same as esCombination but with other constraints
-- on the base ring of the spray.
esCombination' :: forall a. (Eq a, C Rational a, C a) => Spray a -> Map Partition a
-- | Symmetric polynomial as a linear combination of Schur polynomials.
-- Symmetry is not checked.
schurCombination :: forall a. (Eq a, C a) => Spray a -> Map Partition a
-- | Symmetric polynomial as a linear combination of Schur polynomials.
-- Same as schurCombination but with other constraints on the
-- base ring of the spray.
schurCombination' :: forall a. (Eq a, C Rational a, C a) => Spray a -> Map Partition a
-- | Symmetric polynomial as a linear combination of Jack polynomials with
-- a given Jack parameter. Symmetry is not checked.
jackCombination :: (Eq a, C a) => a -> Char -> Spray a -> Map Partition a
-- | Symmetric polynomial as a linear combination of Jack polynomials with
-- symbolic parameter. Symmetry is not checked.
jackSymbolicCombination :: Char -> QSpray -> Map Partition RatioOfQSprays
-- | Symmetric parametric polynomial as a linear combination of Jack
-- polynomials with symbolic parameter. Similar to
-- jackSymbolicCombination but for a parametric spray.
jackSymbolicCombination' :: (Eq a, C a) => Char -> ParametricSpray a -> Map Partition (RatioOfSprays a)
-- | Prints a symmetric spray as a linear combination of monomial symmetric
-- polynomials
--
--
-- >>> putStrLn $ prettySymmetricNumSpray $ schurPol' 3 [3, 1, 1]
-- M[3,1,1] + M[2,2,1]
--
prettySymmetricNumSpray :: (Num a, Ord a, Show a, C a) => Spray a -> String
-- | Prints a symmetric spray as a linear combination of monomial symmetric
-- polynomials
--
--
-- >>> putStrLn $ prettySymmetricQSpray $ jackPol' 3 [3, 1, 1] 2 'J'
-- 42*M[3,1,1] + 28*M[2,2,1]
--
prettySymmetricQSpray :: QSpray -> String
-- | Same as prettySymmetricQSpray but for a QSpray'
-- symmetric spray
prettySymmetricQSpray' :: QSpray' -> String
-- | Prints a symmetric parametric spray as a linear combination of
-- monomial symmetric polynomials
--
--
-- >>> putStrLn $ prettySymmetricParametricQSpray ["a"] $ jackSymbolicPol' 3 [3, 1, 1] 'J'
-- { [ 4*a^2 + 10*a + 6 ] }*M[3,1,1] + { [ 8*a + 12 ] }*M[2,2,1]
--
prettySymmetricParametricQSpray :: [String] -> ParametricQSpray -> String
-- | Laplace-Beltrami operator on the space of homogeneous symmetric
-- polynomials; neither symmetry and homogeneity are checked
laplaceBeltrami :: (Eq a, C a) => a -> Spray a -> Spray a
-- | Calogero-Sutherland operator on the space of homogeneous symmetric
-- polynomials; neither symmetry and homogeneity are checked
calogeroSutherland :: (Eq a, C a) => a -> Spray a -> Spray a
-- | Hall inner product with parameter, aka Jack-scalar product. It makes
-- sense only for symmetric sprays, and the symmetry is not checked.
hallInnerProduct :: forall a. (Eq a, C a) => Spray a -> Spray a -> a -> a
-- | Hall inner product with parameter. Same as hallInnerProduct
-- but with other constraints on the base ring of the sprays.
hallInnerProduct' :: forall a. (Eq a, C Rational a, C a) => Spray a -> Spray a -> a -> a
-- | Hall inner product with parameter. Same as hallInnerProduct
-- but with other constraints on the base ring of the sprays. It is
-- applicable to Spray Int sprays.
hallInnerProduct'' :: forall a. Real a => Spray a -> Spray a -> a -> Rational
-- | Hall inner product with parameter for parametric sprays, because the
-- type of the parameter in hallInnerProduct is strange. For
-- example, a ParametricQSpray spray is a Spray
-- RatioOfQSprays spray, and it makes more sense to compute the Hall
-- product with a Rational parameter then to compute the Hall
-- product with a RatioOfQSprays parameter.
--
--
-- >>> import Math.Algebra.Jack.SymmetricPolynomials
--
-- >>> import Math.Algebra.JackSymbolicPol
--
-- >>> import Math.Algebra.Hspray
--
-- >>> jp = jackSymbolicPol 3 [2, 1] 'P'
--
-- >>> hallInnerProduct''' jp jp 5 == hallInnerProduct jp jp (constantRatioOfSprays 5)
--
hallInnerProduct''' :: forall b. (Eq b, C b, C (BaseRing b) b) => Spray b -> Spray b -> BaseRing b -> b
-- | Hall inner product with parameter for parametric sprays. Same as
-- hallInnerProduct''' but with other constraints on the types.
-- It is applicable to SimpleParametricQSpray sprays, while
-- hallInnerProduct''' is not.
hallInnerProduct'''' :: forall b. (Eq b, C b, C Rational b, C (BaseRing b) b) => Spray b -> Spray b -> BaseRing b -> b
-- | Hall inner product with symbolic parameter. See README for some
-- examples.
symbolicHallInnerProduct :: (Eq a, C a) => Spray a -> Spray a -> Spray a
-- | Hall inner product with symbolic parameter. Same as
-- symbolicHallInnerProduct but with other type constraints.
symbolicHallInnerProduct' :: (Eq a, C Rational (Spray a), C a) => Spray a -> Spray a -> Spray a
-- | Hall inner product with symbolic parameter. Same as
-- symbolicHallInnerProduct but with other type constraints. It
-- is applicable to Spray Int sprays.
symbolicHallInnerProduct'' :: forall a. Real a => Spray a -> Spray a -> QSpray
-- | Kostka numbers <math> for a given weight of the partitions
-- <math> and <math> and a given parameter <math>
-- (these are the standard Kostka numbers when <math>). This
-- returns a map whose keys represent the partitions <math> and the
-- value attached to a partition <math> represents the map
-- <math> where the partition <math> is included in the keys
-- of this map if and only if <math>.
kostkaNumbers :: Int -> Rational -> Map Partition (Map Partition Rational)
-- | Kostka numbers <math> with symbolic parameter <math> for a
-- given weight of the partitions <math> and <math>. This
-- returns a map whose keys represent the partitions <math> and the
-- value attached to a partition <math> represents the map
-- <math> where the partition <math> is included in the keys
-- of this map if and only if <math>.
symbolicKostkaNumbers :: Int -> Map Partition (Map Partition RatioOfQSprays)