-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Jack, zonal, Schur and skew Schur polynomials
--   
--   This library can evaluate Jack polynomials, zonal polynomials, Schur
--   and skew Schur polynomials. It is also able to compute them in
--   symbolic form.
@package jackpolynomials
@version 1.4.0.0


-- | Evaluation of Jack polynomials, zonal polynomials, Schur polynomials
--   and skew Schur polynomials. See README for examples and references.
module Math.Algebra.Jack

-- | 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


-- | 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 motivation of this module.
module Math.Algebra.Jack.SymmetricPolynomials

-- | Checks whether a spray defines a symmetric polynomial; this is useless
--   for Jack polynomials because they always are symmetric, but this
--   module contains everything needed to build this function which can be
--   useful in another context
isSymmetricSpray :: (C a, Eq a) => Spray a -> Bool

-- | Monomial symmetric polynomials
--   
--   <pre>
--   &gt;&gt;&gt; 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
--   </pre>
msPolynomial :: (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

-- | Prints a symmetric spray as a linear combination of monomial symmetric
--   polynomials
--   
--   <pre>
--   &gt;&gt;&gt; putStrLn $ prettySymmetricNumSpray $ schurPol' 3 [3, 1, 1]
--   M[3,1,1] + M[2,2,1]
--   </pre>
prettySymmetricNumSpray :: (Num a, Ord a, Show a, C a) => Spray a -> String

-- | Prints a symmetric spray as a linear combination of monomial symmetric
--   polynomials
--   
--   <pre>
--   &gt;&gt;&gt; putStrLn $ prettySymmetricQSpray $ jackPol' 3 [3, 1, 1] 2 'J'
--   42*M[3,1,1] + 28*M[2,2,1]
--   </pre>
prettySymmetricQSpray :: QSpray -> String

-- | Same as <a>prettySymmetricQSpray</a> but for a <a>QSpray'</a>
--   symmetric spray
prettySymmetricQSpray' :: QSpray' -> String

-- | Prints a symmetric parametric spray as a linear combination of
--   monomial symmetric polynomials
--   
--   <pre>
--   &gt;&gt;&gt; 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]
--   </pre>
prettySymmetricParametricQSpray :: [String] -> ParametricQSpray -> String
laplaceBeltrami :: (Eq a, C a) => a -> Spray a -> Spray a
calogeroSutherland :: (Eq a, C a) => a -> Spray a -> Spray 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. (Ord a, C a) => Int -> Partition -> Spray a

-- | Symbolic skew Schur polynomial
skewSchurPol :: forall a. (Ord 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