-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/
-- | Jack, zonal, Schur, and other symmetric polynomials
--
-- This library can compute Jack polynomials, zonal polynomials, Schur
-- polynomials, flagged Schur polynomials, factorial Schur polynomials,
-- t-Schur polynomials, Hall-Littlewood polynomials, Macdonald
-- polynomials, Kostka-Foulkes polynomials, and Kostka-Macdonald
-- polynomials.
@package jackpolynomials
@version 1.4.5.0
-- | 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
-- | Jack polynomial
jackPol' :: Int -> Partition -> Rational -> Char -> QSpray
-- | Skew Jack polynomial
skewJackPol' :: Int -> Partition -> Partition -> Rational -> Char -> QSpray
-- | Zonal polynomial
zonalPol' :: Int -> Partition -> QSpray
-- | Skew zonal polynomial
skewZonalPol' :: Int -> Partition -> Partition -> QSpray
-- | Schur polynomial
schurPol' :: Int -> Partition -> QSpray
-- | Skew Schur polynomial
skewSchurPol' :: Int -> Partition -> Partition -> QSpray
-- | Jack polynomial
jackPol :: forall a. (Eq a, C a) => Int -> Partition -> a -> Char -> Spray a
-- | Skew Jack polynomial
skewJackPol :: (Eq a, C a) => Int -> Partition -> Partition -> a -> Char -> Spray a
-- | Zonal polynomial
zonalPol :: (Eq a, C a) => Int -> Partition -> Spray a
-- | Zonal polynomial
skewZonalPol :: (Eq a, C a) => Int -> Partition -> Partition -> Spray a
-- | Schur polynomial
schurPol :: forall a. (Eq a, C a) => Int -> Partition -> Spray a
-- | 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
skewJackSymbolicPol :: (Eq a, C a) => Int -> Partition -> Partition -> Char -> ParametricSpray a
skewJackSymbolicPol' :: Int -> Partition -> 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 :: (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' :: (Eq a, C Rational a, C a) => Spray a -> Map Partition a
-- | Symmetric parametric spray as a linear combination of power sum
-- polynomials.
psCombination'' :: (FunctionLike b, Eq b, C b, C (BaseRing b)) => Spray b -> Map Partition b
-- | Symmetric polynomial as a linear combination of complete symmetric
-- homogeneous polynomials. Symmetry is not checked.
cshCombination :: (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' :: (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 :: (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' :: (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 :: (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' :: (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
-- | Prints a symmetric simple parametric spray as a linear combination of
-- monomial symmetric polynomials.
prettySymmetricSimpleParametricQSpray :: [String] -> SimpleParametricQSpray -> 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 Jack parameter, aka Jack-scalar product. It
-- makes sense only for symmetric sprays, and the symmetry is not
-- checked.
hallInnerProduct :: (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' :: (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''' :: (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'''' :: (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 Jack 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 Jack 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)
-- | Skew Kostka numbers <math> with a given Jack parameter
-- <math> and a given skew partition <math>. This returns a
-- map whose keys represent the partitions <math>.
skewKostkaNumbers :: Rational -> Partition -> Partition -> Map Partition Rational
-- | Skew Kostka numbers <math> with symbolic Jack parameter
-- <math> for a given skew partition <math>. This returns a
-- map whose keys represent the partitions <math>.
symbolicSkewKostkaNumbers :: Partition -> Partition -> Map Partition RatioOfQSprays
-- | Kostka-Foulkes polynomial of two given partitions. This is a
-- univariate polynomial whose value at 1 is the Kostka number
-- of the two partitions.
kostkaFoulkesPolynomial :: (Eq a, C a) => Partition -> Partition -> Spray a
-- | Kostka-Foulkes polynomial of two given partitions. This is a
-- univariate polynomial whose value at 1 is the Kostka number
-- of the two partitions.
kostkaFoulkesPolynomial' :: Partition -> Partition -> QSpray
-- | Skew Kostka-Foulkes polynomial. This is a univariate polynomial
-- associated to a skew partition and a partition, and its value at
-- 1 is the skew Kostka number associated to these partitions.
skewKostkaFoulkesPolynomial :: (Eq a, C a) => Partition -> Partition -> Partition -> Spray a
-- | Skew Kostka-Foulkes polynomial. This is a univariate polynomial
-- associated to a skew partition and a partition, and its value at
-- 1 is the skew Kostka number associated to these partitions.
skewKostkaFoulkesPolynomial' :: Partition -> Partition -> Partition -> QSpray
-- | qt-Kostka polynomials, aka Kostka-Macdonald polynomials. They are
-- usually denoted by <math> for two integer partitions
-- <math> and <math>. For a given partition <math>, the
-- function returns the polynomials <math> for all partitions
-- <math> of the same weight as <math>.
qtKostkaPolynomials :: (Eq a, C a) => Partition -> Map Partition (Spray a)
-- | qt-Kostka polynomials, aka Kostka-Macdonald polynomials. They are
-- usually denoted by <math> for two integer partitions
-- <math> and <math>. For a given partition <math>, the
-- function returns the polynomials <math> for all partitions
-- <math> of the same weight as <math>.
qtKostkaPolynomials' :: Partition -> Map Partition QSpray
-- | Skew qt-Kostka polynomials. They are usually denoted by <math>
-- for two integer partitions <math> and <math> defining a
-- skew partition and an integer partition <math>. For given
-- partitions <math> and <math>, the function returns the
-- polynomials <math> for all partitions <math> of the same
-- weight as the skew partition.
qtSkewKostkaPolynomials :: (Eq a, C a) => Partition -> Partition -> Map Partition (Spray a)
-- | Skew qt-Kostka polynomials. They are usually denoted by <math>
-- for two integer partitions <math> and <math> defining a
-- skew partition and an integer partition <math>. For given
-- partitions <math> and <math>, the function returns the
-- polynomials <math> for all partitions <math> of the same
-- weight as the skew partition.
qtSkewKostkaPolynomials' :: Partition -> Partition -> Map Partition QSpray
-- | Hall-Littlewood polynomial of a given partition. This is a
-- multivariate symmetric polynomial whose coefficients are polynomial in
-- a single parameter.
hallLittlewoodPolynomial :: (Eq a, C a) => Int -> Partition -> Char -> SimpleParametricSpray a
-- | Hall-Littlewood polynomial of a given partition. This is a
-- multivariate symmetric polynomial whose coefficients are polynomial in
-- a single parameter.
hallLittlewoodPolynomial' :: Int -> Partition -> Char -> SimpleParametricQSpray
-- | Hall-Littlewood polynomials as linear combinations of Schur
-- polynomials.
transitionsSchurToHallLittlewood :: Int -> Char -> Map Partition (Map Partition (Spray Int))
-- | Skew Hall-Littlewood polynomial of a given skew partition. This is a
-- multivariate symmetric polynomial whose coefficients are polynomial in
-- a single parameter.
skewHallLittlewoodPolynomial :: (Eq a, C a) => Int -> Partition -> Partition -> Char -> SimpleParametricSpray a
-- | Skew Hall-Littlewood polynomial of a given skew partition. This is a
-- multivariate symmetric polynomial whose coefficients are polynomial in
-- one parameter.
skewHallLittlewoodPolynomial' :: Int -> Partition -> Partition -> Char -> SimpleParametricQSpray
-- | t-Schur polynomial. This is a multivariate symmetric polynomial whose
-- coefficients are polynomial in a single parameter.
tSchurPolynomial :: (Eq a, C a) => Int -> Partition -> SimpleParametricSpray a
-- | t-Schur polynomial. This is a multivariate symmetric polynomial whose
-- coefficients are polynomial in one parameter.
tSchurPolynomial' :: Int -> Partition -> SimpleParametricQSpray
-- | Skew t-Schur polynomial of a given skew partition. This is a
-- multivariate symmetric polynomial whose coefficients are polynomial in
-- one parameter.
tSkewSchurPolynomial :: (Eq a, C a) => Int -> Partition -> Partition -> SimpleParametricSpray a
-- | Skew t-Schur polynomial of a given skew partition. This is a
-- multivariate symmetric polynomial whose coefficients are polynomial in
-- one parameter.
tSkewSchurPolynomial' :: Int -> Partition -> Partition -> SimpleParametricQSpray
-- | Macdonald polynomial. This is a symmetric multivariate polynomial
-- depending on two parameters usually denoted by q and
-- t.
--
--
-- >>> macPoly = macdonaldPolynomial 3 [2, 1] 'P'
--
-- >>> putStrLn $ prettySymmetricParametricQSpray ["q", "t"] macPoly
-- { [ 1 ] }*M[2,1] + { [ 2*q.t^2 - q.t - q + t^2 + t - 2 ] %//% [ q.t^2 - 1 ] }*M[1,1,1]
--
macdonaldPolynomial :: (Eq a, C a) => Int -> Partition -> Char -> ParametricSpray a
-- | Macdonald polynomial. This is a symmetric multivariate polynomial
-- depending on two parameters usually denoted by q and
-- t.
macdonaldPolynomial' :: Int -> Partition -> Char -> ParametricQSpray
-- | Skew Macdonald polynomial of a given skew partition. This is a
-- multivariate symmetric polynomial with two parameters.
skewMacdonaldPolynomial :: (Eq a, C a) => Int -> Partition -> Partition -> Char -> ParametricSpray a
-- | Skew Macdonald polynomial of a given skew partition. This is a
-- multivariate symmetric polynomial with two parameters.
skewMacdonaldPolynomial' :: Int -> Partition -> Partition -> Char -> ParametricQSpray
-- | Macdonald J-polynomial. This is a multivariate symmetric polynomial
-- whose coefficients are polynomial in two parameters.
macdonaldJpolynomial :: forall a. (Eq a, C a) => Int -> Partition -> SimpleParametricSpray a
-- | Macdonald J-polynomial. This is a multivariate symmetric polynomial
-- whose coefficients are polynomial in two parameters.
macdonaldJpolynomial' :: Int -> Partition -> SimpleParametricQSpray
-- | Skew Macdonald J-polynomial. This is a multivariate symmetric
-- polynomial whose coefficients depend on two parameters.
skewMacdonaldJpolynomial :: (Eq a, C a) => Int -> Partition -> Partition -> ParametricSpray a
-- | Skew Macdonald J-polynomial. This is a multivariate symmetric
-- polynomial whose coefficients depend on two parameters.
skewMacdonaldJpolynomial' :: Int -> Partition -> Partition -> ParametricQSpray
-- | Modified Macdonald polynomial. This is a multivariate symmetric
-- polynomial whose coefficients are polynomials in two parameters.
modifiedMacdonaldPolynomial :: (Eq a, C a) => Int -> Partition -> SimpleParametricSpray a
-- | Modified Macdonald polynomial. This is a multivariate symmetric
-- polynomial whose coefficients are polynomials in two parameters.
modifiedMacdonaldPolynomial' :: Int -> Partition -> SimpleParametricQSpray
-- | Flagged Schur polynomial. A flagged Schur polynomial is not symmetric
-- in general.
flaggedSchurPol :: (Eq a, C a) => Partition -> [Int] -> [Int] -> Spray a
-- | Flagged Schur polynomial. A flagged Schur polynomial is not symmetric
-- in general.
flaggedSchurPol' :: Partition -> [Int] -> [Int] -> QSpray
-- | Flagged skew Schur polynomial. A flagged skew Schur polynomial is not
-- symmetric in general.
flaggedSkewSchurPol :: (Eq a, C a) => Partition -> Partition -> [Int] -> [Int] -> Spray a
-- | Flagged skew Schur polynomial. A flagged skew Schur polynomial is not
-- symmetric in general.
flaggedSkewSchurPol' :: Partition -> Partition -> [Int] -> [Int] -> QSpray
-- | Factorial Schur polynomial. See Kreiman's paper Products of
-- factorial Schur functions for the definition.
factorialSchurPol :: (Eq a, C a) => Int -> Partition -> [a] -> Spray a
-- | Factorial Schur polynomial. See Kreiman's paper Products of
-- factorial Schur functions for the definition.
factorialSchurPol' :: Int -> Partition -> [Rational] -> QSpray
-- | Skew factorial Schur polynomial. See Macdonald's paper Schur
-- functions: theme and variations, 6th variation, for the
-- definition.
skewFactorialSchurPol :: (Eq a, C a) => Int -> Partition -> Partition -> IntMap a -> Spray a
-- | Skew factorial Schur polynomial. See Macdonald's paper Schur
-- functions: theme and variations, 6th variation, for the
-- definition.
skewFactorialSchurPol' :: Int -> Partition -> Partition -> IntMap Rational -> QSpray