Math.Algebra.SymmetricPolynomials

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.

Checking symmetry

isSymmetricSpray :: (C a, Eq a) => Spray a -> Bool

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

Classical symmetric polynomials

msPolynomial

Arguments

:: (C a, Eq a)

=> Int

number of variables

-> Partition

integer partition

-> Spray a

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

psPolynomial

Arguments

:: (C a, Eq a)

=> Int

number of variables

-> Partition

integer 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

cshPolynomial

Arguments

:: (C a, Eq a)

=> Int

number of variables

-> Partition

integer 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

esPolynomial

Arguments

:: (C a, Eq a)

=> Int

number of variables

-> Partition

integer 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

Decomposition of symmetric polynomials

msCombination :: C a => Spray a -> Map Partition a

Symmetric polynomial as a linear combination of monomial symmetric polynomials.

psCombination :: forall a. (Eq a, 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 Rational 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.

cshCombination :: forall a. (Eq 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 Rational 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.

esCombination :: forall a. (Eq 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 Rational 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.

schurCombination :: forall a. (Eq 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 Rational 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.

jackCombination

Arguments

:: (Eq a, C a)

=> a

Jack parameter

-> Char

which Jack polynomials, J, C, P or Q

-> Spray a

spray representing a symmetric polynomial

-> Map Partition a

map representing the linear combination; a partition lambda in the keys of this map corresponds to the term coeff *^ jackPol' n lambda alpha which, where coeff is the value attached to this key and n is the number of variables of the spray

Symmetric polynomial as a linear combination of Jack polynomials with a given Jack parameter. Symmetry is not checked.

jackSymbolicCombination

Arguments

:: Char

which Jack polynomials, J, C, P or Q

-> QSpray

spray representing a symmetric polynomial

-> Map Partition RatioOfQSprays

map representing the linear combination; a partition lambda in the keys of this map corresponds to the term coeff *^ jackSymbolicPol' n lambda which, where coeff is the value attached to this key and n is the number of variables of the spray

Symmetric polynomial as a linear combination of Jack polynomials with symbolic parameter. Symmetry is not checked.

jackSymbolicCombination'

Arguments

:: (Eq a, C a)

=> Char

which Jack polynomials, J, C, P or Q

-> ParametricSpray a

parametric spray representing a symmetric polynomial

-> Map Partition (RatioOfSprays a)

map representing the linear combination; a partition lambda in the keys of this map corresponds to the term coeff *^ jackSymbolicPol' n lambda which, where coeff is the value attached to this key and n is the number of variables of the spray

Symmetric parametric polynomial as a linear combination of Jack polynomials with symbolic parameter. Similar to jackSymbolicCombination but for a parametric spray.

Printing symmetric polynomials

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 $ prettySymmetricNumSpray $ schurPol' 3 [3, 1, 1]
M[3,1,1] + M[2,2,1]

prettySymmetricQSpray :: QSpray -> 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

prettySymmetricParametricQSpray :: [String] -> ParametricQSpray -> 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]

prettySymmetricSimpleParametricQSpray :: [String] -> SimpleParametricQSpray -> String

Prints a symmetric simple parametric spray as a linear combination of monomial symmetric polynomials.

Operators on the space of symmetric polynomials

laplaceBeltrami :: (Eq a, C a) => a -> Spray a -> Spray a

Laplace-Beltrami operator on the space of homogeneous symmetric polynomials; neither symmetry and homogeneity are checked.

calogeroSutherland :: (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

Hall inner product of symmetric polynomials

hallInnerProduct

Arguments

:: forall a. (Eq a, C a)

=> Spray a

spray

-> Spray a

spray

-> a

parameter

-> a

Hall inner product with parameter, aka Jack-scalar product. It makes sense only for symmetric sprays, and the symmetry is not checked.

hallInnerProduct'

Arguments

:: forall a. (Eq a, C Rational a, C a)

=> Spray a

spray

-> Spray a

spray

-> a

parameter

-> a

Hall inner product with parameter. Same as hallInnerProduct but with other constraints on the base ring of the sprays.

hallInnerProduct''

Arguments

:: forall a. Real a

=> Spray a

spray

-> Spray a

spray

-> a

parameter

-> Rational

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'''

Arguments

:: forall b. (Eq b, C b, C (BaseRing b) b)

=> Spray b

parametric spray

-> Spray b

parametric spray

-> BaseRing b

parameter

-> b

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''''

Arguments

:: forall b. (Eq b, C b, C Rational b, C (BaseRing b) b)

=> Spray b

parametric spray

-> Spray b

parametric spray

-> BaseRing b

parameter

-> 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.

symbolicHallInnerProduct :: (Eq a, C a) => Spray a -> Spray a -> Spray a

Hall inner product with symbolic parameter. See README for some examples.

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.

symbolicHallInnerProduct'' :: forall a. Real a => Spray a -> Spray a -> QSpray

Hall inner product with symbolic parameter. Same as symbolicHallInnerProduct but with other type constraints. It is applicable to Spray Int sprays.

Kostka numbers

kostkaNumbers

Arguments

:: Int

weight of the partitions

-> Rational

Jack parameter

-> Map Partition (Map Partition Rational)

Kostka numbers \(K_{\lambda,\mu}(\alpha)\) for a given weight of the partitions \(\lambda\) and \(\mu\) and a given parameter \(\alpha\) (these are the standard Kostka numbers when \(\alpha=1\)). This returns a map whose keys represent the partitions \(\lambda\) and the value attached to a partition \(\lambda\) represents the map \(\mu \mapsto K_{\lambda,\mu}(\alpha)\) where the partition \(\mu\) is included in the keys of this map if and only if \(K_{\lambda,\mu}(\alpha) \neq 0\).

symbolicKostkaNumbers :: Int -> Map Partition (Map Partition RatioOfQSprays)

Kostka numbers \(K_{\lambda,\mu}(\alpha)\) with symbolic parameter \(\alpha\) for a given weight of the partitions \(\lambda\) and \(\mu\). This returns a map whose keys represent the partitions \(\lambda\) and the value attached to a partition \(\lambda\) represents the map \(\mu \mapsto K_{\lambda,\mu}(\alpha)\) where the partition \(\mu\) is included in the keys of this map if and only if \(K_{\lambda,\mu}(\alpha) \neq 0\).

Kostka-Foulkes polynomials

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

Kostka-Foulkes polynomial of two given partitions. This is a univariate polynomial whose value at 1 is the Kostka number of the two partitions.

Hall-Littlewood polynomials

hallLittlewoodPolynomial

Arguments

:: (Eq a, C a)

=> Int

number of variables

-> Partition

integer partition

-> Char

which Hall-Littlewood polynomial, P or Q

-> SimpleParametricSpray a

Hall-Littlewood polynomial of a given partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in one parameter.

hallLittlewoodPolynomial'

Arguments

:: Int

number of variables

-> Partition

integer partition

-> Char

which Hall-Littlewood polynomial, P or Q

-> SimpleParametricQSpray

Hall-Littlewood polynomial of a given partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in one parameter.

transitionsSchurToHallLittlewood

Arguments

:: Int

weight of the partitions of the Hall-Littlewood polynomials

-> Char

which Hall-Littlewood polynomials, P or Q

-> Map Partition (Map Partition (Spray Int))

Hall-Littlewood polynomials as linear combinations of Schur polynomials.

skewHallLittlewoodPolynomial

Arguments

:: (Eq a, C a)

=> Int

number of variables

-> Partition

outer partition of the skew partition

-> Partition

inner partition of the skew partition

-> Char

which skew Hall-Littlewood polynomial, P or Q

-> 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'

Arguments

:: Int

number of variables

-> Partition

outer partition of the skew partition

-> Partition

inner partition of the skew partition

-> Char

which skew Hall-Littlewood polynomial, P or Q

-> SimpleParametricQSpray

Skew Hall-Littlewood polynomial of a given skew partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in one parameter.

Flagged Schur polynomials

flaggedSchurPol

Arguments

:: (Eq a, C a)

=> Partition

integer partition

-> [Int]

lower bounds

-> [Int]

upper bounds

-> Spray a

Flagged Schur polynomial. A flagged Schur polynomial is not symmetric in general.

flaggedSchurPol'

Arguments

:: Partition

integer partition

-> [Int]

lower bounds

-> [Int]

upper bounds

-> QSpray

Flagged Schur polynomial. A flagged Schur polynomial is not symmetric in general.

flaggedSkewSchurPol

Arguments

:: (Eq a, C a)

=> Partition

outer partition of the skew partition

-> Partition

inner partition of the skew partition

-> [Int]

lower bounds

-> [Int]

upper bounds

-> Spray a

Flagged skew Schur polynomial. A flagged skew Schur polynomial is not symmetric in general.

flaggedSkewSchurPol'

Arguments

:: Partition

outer partition of the skew partition

-> Partition

inner partition of the skew partition

-> [Int]

lower bounds

-> [Int]

upper bounds

-> QSpray

Flagged skew Schur polynomial. A flagged skew Schur polynomial is not symmetric in general.

Factorial Schur polynomials

factorialSchurPol

Arguments

:: (Eq a, C a)

=> Int

number of variables

-> Partition

integer partition

-> [a]

the sequence denoted by \(y\) in the reference paper

-> Spray a

Factorial Schur polynomial. See Kreiman's paper Products of factorial Schur functions for the definition.

factorialSchurPol'

Arguments

:: Int

number of variables

-> Partition

integer partition

-> [Rational]

the sequence denoted by \(y\) in the reference paper

-> QSpray

Factorial Schur polynomial. See Kreiman's paper Products of factorial Schur functions for the definition.

skewFactorialSchurPol

Arguments

:: (Eq a, C a)

=> Int

number of variables

-> Partition

outer partition of the skew partition

-> Partition

inner partition of the skew partition

-> IntMap a

the sequence denoted by \(a\) in the reference paper

-> Spray a

Skew factorial Schur polynomial. See Macdonald's paper Schur functions: theme and variations, 6th variation, for the definition.

skewFactorialSchurPol'

Arguments

:: Int

number of variables

-> Partition

outer partition of the skew partition

-> Partition

inner partition of the skew partition

-> IntMap Rational

the sequence denoted by \(a\) in the reference paper

-> QSpray

Skew factorial Schur polynomial. See Macdonald's paper Schur functions: theme and variations, 6th variation, for the definition.