h*<;*5      !"#$%&'()*+,-./012341.4.2.0 Safe-Inferred56789:;<=>?@ABCDEFGEvaluation of Jack polynomials.(c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-InferredjackpolynomialsEvaluation of Jack polynomialjackpolynomialsEvaluation of Jack polynomialjackpolynomialsEvaluation of zonal polynomialjackpolynomialsEvaluation of zonal polynomialjackpolynomialsEvaluation of Schur polynomialjackpolynomialsEvaluation of Schur polynomialjackpolynomials%Evaluation of a skew Schur polynomialjackpolynomials%Evaluation of a skew Schur polynomialjackpolynomialsvalues of the variablesjackpolynomialspartition of integersjackpolynomialsJack parameterjackpolynomialswhich Jack polynomial, J, C, P or Qjackpolynomialsvalues of the variablesjackpolynomialspartition of integersjackpolynomialsJack parameterjackpolynomialswhich Jack polynomial, J, C, P or Qjackpolynomialsvalues of the variablesjackpolynomialspartition of integersjackpolynomialsvalues of the variablesjackpolynomialspartition of integersjackpolynomialsvalues of the variablesjackpolynomialspartition of integers jackpolynomialsvalues of the variablesjackpolynomialspartition of integers jackpolynomialsvalues of the variablesjackpolynomials)the outer partition of the skew partitionjackpolynomials)the inner partition of the skew partitionjackpolynomialsvalues of the variablesjackpolynomials)the outer partition of the skew partitionjackpolynomials)the inner partition of the skew partition   Safe-Inferreda jackpolynomialsInefficient hypergeometric function of a matrix argument (for testing purpose)  Symbolic Jack polynomials.(c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-Inferred jackpolynomialsSymbolic Jack polynomial jackpolynomialsSymbolic Jack polynomial jackpolynomialsSymbolic zonal polynomial jackpolynomialsSymbolic zonal polynomialjackpolynomialsSymbolic Schur polynomialjackpolynomialsSymbolic Schur polynomialjackpolynomialsSymbolic skew Schur polynomialjackpolynomialsSymbolic skew Schur polynomial jackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsJack parameterjackpolynomialswhich Jack polynomial, J, C, P or Q jackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsJack parameterjackpolynomialswhich Jack polynomial, J, C, P or Q jackpolynomialsnumber of variablesjackpolynomialspartition of integers jackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partition    .Jack polynomials with symbolic Jack parameter.(c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-Inferredjackpolynomials,Jack polynomial with symbolic Jack parameterjackpolynomials,Jack polynomial with symbolic Jack parameterjackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialswhich Jack polynomial, J, C, P or Qjackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialswhich Jack polynomial, J, C, P or Q$Some utilities for Jack polynomials.(c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-Inferred:1Hjackpolynomialsmonomial symmetric polynomialjackpolynomialsMonomial 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.x3^2jackpolynomials6Checks 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:1spray = schurPol' 4 [2, 2] ^+^ schurPol' 3 [2, 1]isSymmetricSpray sprayjackpolynomialsSymmetric polynomial as a linear combination of monomial symmetric polynomials.jackpolynomialsPrints 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]jackpolynomialsPrints 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]jackpolynomialsSame as  but for a I symmetric sprayjackpolynomialsPrints a symmetric parametric spray as a linear combination of monomial symmetric polynomialsputStrLn $ 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]jackpolynomialsLaplace-Beltrami operator on the space of homogeneous symmetric polynomials; neither symmetry and homogeneity are checkedjackpolynomialsCalogero-Sutherland operator on the space of homogeneous symmetric polynomials; neither symmetry and homogeneity are checkedjackpolynomialsPower 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^3Jjackpolynomialsmonomial symmetric polynomial as a linear combination of power sum polynomialsKjackpolynomials$the factor in the Hall inner productLjackpolynomialssymmetric polynomial as a linear combination of power sum polynomialsjackpolynomialsSymmetric polynomial as a linear combination of power sum polynomials. Symmetry is not checked.jackpolynomialsSymmetric polynomial as a linear combination of power sum polynomials. Same as  psCombination: but with other constraints on the base ring of the spray.Mjackpolynomials%the Hall inner product with parameter jackpolynomialsHall inner product with parameter, aka Jack-scalar product. It makes sense only for symmetric sprays, and the symmetry is not checked. !jackpolynomials+Hall inner product with parameter. Same as hallInnerProduct= but with other constraints on the base ring of the sprays."jackpolynomials+Hall inner product with parameter. Same as hallInnerProduct but with other constraints on the base ring of the sprays. It is applicable to  Spray Int sprays.#jackpolynomialsHall inner product with parameter for parametric sprays, because the type of the parameter in hallInnerProduct is strange. For example, a ParametricQSpray spray is a Spray RatioOfQSprays spray, and it makes more sense to compute the Hall product with a Rational5 parameter then to compute the Hall product with a RatioOfQSprays parameter.-import Math.Algebra.Jack.SymmetricPolynomials#import Math.Algebra.JackSymbolicPolimport Math.Algebra.Hspray!jp = jackSymbolicPol 3 [2, 1] 'P'hallInnerProduct''' jp jp 5 == hallInnerProduct jp jp (constantRatioOfSprays 5)$jackpolynomialsHall 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.Njackpolynomials.the Hall inner product with symbolic parameter%jackpolynomialsHall inner product with symbolic parameter. See README for some examples.&jackpolynomials4Hall inner product with symbolic parameter. Same as symbolicHallInnerProduct# but with other type constraints.'jackpolynomials4Hall inner product with symbolic parameter. Same as symbolicHallInnerProduct8 but with other type constraints. It is applicable to  Spray Int sprays.(jackpolynomials)Complete symmetric homogeneous polynomial0putStrLn $ 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^3Ojackpolynomialspower sum polynomial as a linear combination of complete symmetric homogeneous polynomialsPjackpolynomialsmonomial symmetric polynomial as a linear combination of complete symmetric homogeneous polynomialsQjackpolynomialssymmetric polynomial as a linear combination of complete symmetric homogeneous polynomials)jackpolynomialsSymmetric polynomial as a linear combination of complete symmetric homogeneous polynomials. Symmetry is not checked.*jackpolynomialsSymmetric polynomial as a linear combination of complete symmetric homogeneous polynomials. Same as cshCombination: but with other constraints on the base ring of the spray.+jackpolynomials Elementary symmetric polynomial./putStrLn $ prettyQSpray (esPolynomial 3 [2, 1])7x^2.y + x^2.z + x.y^2 + 3*x.y.z + x.z^2 + y^2.z + y.z^2Rjackpolynomialspower sum polynomial as a linear combination of elementary symmetric polynomialsSjackpolynomialsmonomial symmetric polynomial as a linear combination of elementary symmetric polynomialsTjackpolynomialssymmetric polynomial as a linear combination of elementary symmetric polynomials,jackpolynomialsSymmetric polynomial as a linear combination of elementary symmetric polynomials. Symmetry is not checked.-jackpolynomialsSymmetric polynomial as a linear combination of elementary symmetric polynomials. Same as  esCombination: but with other constraints on the base ring of the spray.Ujackpolynomialscomplete symmetric homogeneous polynomial as a linear combination of Schur polynomialsVjackpolynomialssymmetric polynomial as a linear combination of Schur polynomials.jackpolynomialsSymmetric polynomial as a linear combination of Schur polynomials. Symmetry is not checked./jackpolynomialsSymmetric polynomial as a linear combination of Schur polynomials. Same as schurCombination: but with other constraints on the base ring of the spray.0jackpolynomialsKostka 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 \mu6 is included in the keys of this map if and only if K_{\lambda,\mu}(\alpha) \neq 0.1jackpolynomialsKostka 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 \mu6 is included in the keys of this map if and only if K_{\lambda,\mu}(\alpha) \neq 0.Wjackpolynomials8monomial symmetric polynomials in Jack polynomials basisXjackpolynomials8monomial symmetric polynomials in Jack polynomials basis2jackpolynomialsSymmetric polynomial as a linear combination of Jack polynomials with a given Jack parameter. Symmetry is not checked.3jackpolynomialsSymmetric polynomial as a linear combination of Jack polynomials with symbolic parameter. Symmetry is not checked.4jackpolynomialsSymmetric parametric polynomial as a linear combination of Jack polynomials with symbolic parameter. Similar to jackSymbolicCombination but for a parametric spray.Hjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialsnumber of variablesjackpolynomialsinteger partitionMjackpolynomialssprayjackpolynomialssprayjackpolynomials parameter jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter!jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter"jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter#jackpolynomialsparametric sprayjackpolynomialsparametric sprayjackpolynomials parameter$jackpolynomialsparametric sprayjackpolynomialsparametric sprayjackpolynomials parameter(jackpolynomialsnumber of variablesjackpolynomialsinteger partition+jackpolynomialsnumber of variablesjackpolynomialsinteger partition0jackpolynomialsweight of the partitions2jackpolynomialsJack parameterjackpolynomialswhich Jack polynomials, J, C, P or Qjackpolynomials)spray representing a symmetric polynomialjackpolynomials5map representing the linear combination; a partition lambda1 in the keys of this map corresponds to the term &coeff *^ jackPol' n lambda alpha which, where coeff' is the value attached to this key and n( is the number of variables of the spray3jackpolynomialswhich Jack polynomials, J, C, P or Qjackpolynomials)spray representing a symmetric polynomialjackpolynomials5map representing the linear combination; a partition lambda1 in the keys of this map corresponds to the term (coeff *^ jackSymbolicPol' n lambda which, where coeff' is the value attached to this key and n( is the number of variables of the spray4jackpolynomialswhich Jack polynomials, J, C, P or Qjackpolynomials4parametric spray representing a symmetric polynomialjackpolynomials5map representing the linear combination; a partition lambda1 in the keys of this map corresponds to the term (coeff *^ jackSymbolicPol' n lambda which, where coeff' is the value attached to this key and n( is the number of variables of the spray!(+)*,-./234 !"#$%&'01!(+)*,-./234 !"#$%&'01      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`ab.jackpolynomials-1.4.2.0-HknkZs0KsbVEH1D6J0AHUlMath.Algebra.JackMath.Algebra.Jack.HypergeoPQMath.Algebra.JackPolMath.Algebra.JackSymbolicPol!Math.Algebra.SymmetricPolynomialsjackpolynomialsMath.Algebra.Jack.Internal Partitionjack'jackzonal'zonalschur'schur skewSchur' skewSchur hypergeoPQjackPol'jackPol zonalPol'zonalPol schurPol'schurPol skewSchurPol' skewSchurPoljackSymbolicPol'jackSymbolicPol msPolynomialisSymmetricSpray msCombinationprettySymmetricNumSprayprettySymmetricQSprayprettySymmetricQSpray'prettySymmetricParametricQSpraylaplaceBeltramicalogeroSutherland psPolynomial psCombinationpsCombination'hallInnerProducthallInnerProduct'hallInnerProduct''hallInnerProduct'''hallInnerProduct''''symbolicHallInnerProductsymbolicHallInnerProduct'symbolicHallInnerProduct'' cshPolynomialcshCombinationcshCombination' esPolynomial esCombinationesCombination'schurCombinationschurCombination' kostkaNumberssymbolicKostkaNumbersjackCombinationjackSymbolicCombinationjackSymbolicCombination' jackCoeffP jackCoeffQ jackCoeffCjackSymbolicCoeffCjackSymbolicCoeffPinvjackSymbolicCoeffQinv _betaratio_betaRatioOfSprays _isPartition_N_fromIntskewSchurLRCoefficientsisSkewPartition sprayToMap comboToSpray_kostkaNumbers_inverseKostkaMatrix_symbolicKostkaNumbers_inverseSymbolicKostkaMatrixmsPolynomialUnsafe%hspray-0.5.2.0-1Cb7uhS7EN124ie3vUW61pMath.Algebra.HsprayQSpray' mspInPSbasiszlambda_psCombination_hallInnerProduct_symbolicHallInnerProduct pspInCSHbasis mspInCSHbasis_cshCombination pspInESbasis mspInESbasis_esCombinationcshInSchurBasis_schurCombinationmsPolynomialsInJackBasis msPolynomialsInJackSymbolicBasis