-- 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 1.4.1.0


-- | 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. (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


-- | 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; 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 polynomial
--   
--   <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

-- | Power sum polynomial
--   
--   <pre>
--   &gt;&gt;&gt; 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
--   </pre>
psPolynomial :: (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 <tt>psCombination</tt> 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

-- | 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

-- | 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. 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 <tt>hallInnerProduct</tt>
--   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 <tt>hallInnerProduct</tt>
--   but with other constraints on the base ring of the sprays. It is
--   applicable to <tt>Spray Int</tt> 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 <tt>hallInnerProduct</tt> is strange. For
--   example, a <tt>ParametricQSpray</tt> spray is a <tt>Spray
--   RatioOfQSprays</tt> spray, and it makes more sense to compute the Hall
--   product with a <tt>Rational</tt> parameter then to compute the Hall
--   product with a <tt>RatioOfQSprays</tt> parameter.
--   
--   <pre>
--   &gt;&gt;&gt; import Math.Algebra.Jack.SymmetricPolynomials
--   
--   &gt;&gt;&gt; import Math.Algebra.JackSymbolicPol
--   
--   &gt;&gt;&gt; import Math.Algebra.Hspray
--   
--   &gt;&gt;&gt; jp = jackSymbolicPol 3 [2, 1] 'P'
--   
--   &gt;&gt;&gt; hallInnerProduct''' jp jp 5 == hallInnerProduct jp jp (constantRatioOfSprays 5)
--   </pre>
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
--   <tt>hallInnerProduct'''</tt> but with other constraints on the types.
--   It is applicable to <tt>SimpleParametricQSpray</tt> sprays, while
--   <tt>hallInnerProduct'''</tt> 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
--   <tt>symbolicHallInnerProduct</tt> 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
--   <tt>symbolicHallInnerProduct</tt> but with other type constraints. It
--   is applicable to <tt>Spray Int</tt> sprays.
symbolicHallInnerProduct'' :: forall a. Real a => Spray a -> Spray a -> QSpray