jackpolynomials

Jack, zonal, Schur, and other symmetric polynomials.

Schur polynomials have applications in combinatorics and zonal polynomials have applications in multivariate statistics. They are particular cases of Jack polynomials, which are multivariate symmetric polynomials. This package allows to compute these polynomials. It also allows to compute other symmetric polynomials: \(t\)-Schur polynomials, Hall-Littlewood polynomials, and Macdonald polynomials. In addition, it provides some functions to compute Kostka-Jack numbers, Kostka-Foulkes polynomials, Kostka-Macdonald polynomials, Hall polynomials, and to enumerate Gelfand-Tsetlin patterns.

Evaluation of the Jack polynomial with Jack parameter 2, associated to the integer partition [3, 1], at x1 = 1 and x2 = 1:

import Math.Algebra.Jack
jack' [1, 1] [3, 1] 2 'J'
-- 48 % 1

The last argument, here 'J', is used to specify the choice of the Jack polynomial, because there are four possible Jack polynomials for a given Jack parameter and a given integer partition: the \(J\)-polynomial, the \(P\)-polynomial, the \(Q\)-polynomial and the \(C\)-polynomial, each corresponding to a certain normalization.

The non-evaluated Jack polynomial:

import Math.Algebra.JackPol
import Math.Algebra.Hspray
jp = jackPol' 2 [3, 1] 2 'J'
putStrLn $ prettyQSpray jp
-- 18*x^3.y + 12*x^2.y^2 + 18*x.y^3
evalSpray jp [1, 1]
-- 48 % 1

The first argument, here 2, is the number of variables of the polynomial.

Jack polynomials are generalized by skew Jack polynomials, which are available in the package as of version 1.4.5.0.

Symbolic Jack parameter

As of version 1.2.0.0, it is possible to get Jack polynomials with a symbolic Jack parameter:

import Math.Algebra.JackSymbolicPol
import Math.Algebra.Hspray
jp = jackSymbolicPol' 2 [3, 1] 'J'
putStrLn $ prettyParametricQSpray jp
-- { [ 2*a^2 + 4*a + 2 ] }*X^3.Y + { [ 4*a + 4 ] }*X^2.Y^2 + { [ 2*a^2 + 4*a + 2 ] }*X.Y^3
putStrLn $ prettyQSpray' $ substituteParameters jp [2]
-- 18*x^3.y + 12*x^2.y^2 + 18*x.y^3

This is possible thanks to the hspray package which provides the type ParametricSpray. An object of this type represents a multivariate polynomial whose coefficients depend on some parameters which are symbolically treated. The type of the Jack polynomial returned by the jackSymbolicPol function is ParametricSpray a, and it is ParametricQSpray for the jackSymbolicPol' function. The type ParametricQSpray is an alias of ParametricSpray Rational.

From the definition of Jack polynomials, as well as from their implementation in this package, the coefficients of the Jack polynomials are fractions of polynomials in the Jack parameter. However, in the above example, one can see that the coefficients of the Jack polynomial jp are polynomials in the Jack parameter a. This fact actually is always true for the \(J\)-Jack polynomials (not for \(C\), \(P\) and \(Q\)). This is a consequence of the Knop & Sahi combinatorial formula. But be aware that in spite of this fact, the coefficients of the polynomials returned by Haskell are fractions of polynomials, in the sense that this is the nature of the ParametricSpray objects.

Note that if you use the function jackSymbolicPol to get a ParametricSpray Double object in the output, it is not guaranted that you will visually get some polynomials in the Jack parameter for the coefficients, because the arithmetic operations are not exact with the Double type.

Showing symmetric polynomials

As of version 1.2.1.0, there is a module providing some functions to print a symmetric polynomial as a linear combination of the monomial symmetric polynomials. This can considerably shorten the expression of a symmetric polynomial as compared to its expression in the canonical basis, and the motivation to add this module to the package is that any Jack polynomial is a symmetric polynomial. Here is an example:

import Math.Algebra.JackPol
import Math.Algebra.SymmetricPolynomials
jp = jackPol' 3 [3, 1, 1] 2 'J'
putStrLn $ prettySymmetricQSpray jp
-- 42*M[3,1,1] + 28*M[2,2,1]

And another example, involving a Jack polynomial with symbolic Jack parameter:

import Math.Algebra.JackSymbolicPol
import Math.Algebra.SymmetricPolynomials
jp = jackSymbolicPol' 3 [3, 1, 1] 'J'
putStrLn $ prettySymmetricParametricQSpray ["a"] jp
-- { [ 4*a^2 + 10*a + 6 ] }*M[3,1,1] + { [ 8*a + 12 ] }*M[2,2,1]

Of course you can use these functions for other polynomials, but carefully: they do not check the symmetry. This new module provides the function isSymmetricSpray to check the symmetry of a polynomial, much more efficient than the function with the same name in the hspray package.

Hall inner product

As of version 1.4.1.0, the package provides an implementation of the Hall inner product with Jack parameter, aka the Jack scalar product. It is known that the Jack polynomials with Jack parameter \(\alpha\) are orthogonal for the Hall inner product with Jack parameter \(\alpha\).

There is a function hallInnerProduct as well as a function symbolicHallInnerProduct. The latter allows to get the Hall inner product of two symmetric polynomials without substituting a value to the parameter \(\alpha\). The Hall inner product of two symmetric polynomials is a polynomial in \(\alpha\), so the result of symbolicHallInnerProduct is a Spray object.

Let's see a first example with a power sum polynomial. These symmetric polynomials are implemented in the package. We display the result by using alpha to denote the parameter of the Hall product.

import Math.Algebra.SymmetricPolynomials 
import Math.Algebra.Hspray hiding (psPolynomial)
psPoly = psPolynomial 4 [2, 1, 1] :: QSpray
hip = symbolicHallInnerProduct psPoly psPoly
putStrLn $ prettyQSprayXYZ ["alpha"] hip
-- 4*alpha^3

Now let's consider the following situation. We want to get the symbolic Hall inner product of a Jack polynomial with itself, and we deal with a symbolic Jack parameter in this polynomial. We denote it by t to distinguish it from the parameter of the Hall product that we still denote by alpha.

The signature of the symbolicHallInnerProduct is a bit misleading:

Spray a -> Spray a -> Spray a

because the Spray a of the output is not of the same family as the two Spray a inputs: this is a univariate polynomial in \(\alpha\).

We use the function jackSymbolicPol' to compute a Jack polynomial. It returns a ParametricQSpray spray, a type alias of Spray RatioOfQSprays.

import Math.Algebra.JackSymbolicPol
import Math.Algebra.SymmetricPolynomials 
import Math.Algebra.Hspray 
jp = jackSymbolicPol' 2 [3, 1] 'P'
hip = symbolicHallInnerProduct jp jp
putStrLn $ prettyParametricQSprayABCXYZ ["t"] ["alpha"] hip
-- { [ 3*t^2 + 6*t + 11 ] %//% [ t^2 + 2*t + 1 ] }*alpha^2 + { [ 4*t^2 + 16*t + 16 ] %//% [ t^2 + 2*t + 1 ] }*alpha

One could be interested in computing the Hall inner product of a Jack polynomial with itself when the Jack parameter and the parameter of the Hall product are identical. That is, we want to take alpha = t in the above expression. Since the symbolic Hall product is a ParametricQSpray spray, one can substitute its variable alpha by a RatioOfQSprays object. On the other hand, t represents a QSpray object, but one can identify a QSpray to a RatioOfQSprays by taking the unit spray as the denominator, that is, by applying the asRatioOfSprays function. Finally we get the desired result if we evaluate the symbolic Hall product by replacing alpha with asRatioOfSprays (qlone 1), since t is the first polynomial variable, qlone 1.

prettyRatioOfQSpraysXYZ ["t"] $ evaluate hip [asRatioOfSprays (qlone 1)]
-- [ 3*t^4 + 10*t^3 + 27*t^2 + 16*t ] %//% [ t^2 + 2*t + 1 ]

Hall-Littlewood polynomials

The package can also compute the Hall-Littlewood polynomials. A Hall-Littlewood polynomial is a multivariate symmetric polynomial associated to an integer partition and whose coefficients depend on a parameter. More precisely, the coefficients are some polynomials in this parameter. So the Hall-Littlewood polynomials implemented in the package, returned by the hallLittlewoodPolynomial function, are represented by some sprays of type SimpleParametricSpray a, an alias of the type Spray (Spray a).

When the value of the parameter of a Hall-Littlewood \(P\)-polynomial is 0, then this polynomial is the Schur polynomial of the given partition.

import Math.Algebra.JackPol
import Math.Algebra.SymmetricPolynomials 
import Math.Algebra.Hspray 
lambda = [2, 1]
hlPoly = hallLittlewoodPolynomial' 3 lambda 'P' 
putStrLn $ prettySymmetricSimpleParametricQSpray ["t"] hlPoly
-- (1)*M[2,1] + (-t^2 - t + 2)*M[1,1,1]
hlPolyAt0 = substituteParameters hlPoly [0]
hlPolyAt0 == schurPol' 3 lambda
-- True

Macdonald polynomials

As of version 1.4.5.0, the package can compute some Macdonald polynomials. The Macdonald polynomials are symmetric multivariate polynomials whose coefficients depend on two parameters usually denoted by \(q\) and \(t\). Let's consider for example the Macdonald \(P\)-polynomial. It generalizes the Hall-Littlewood \(P\)-polynomial: the Hall-Littlewood \(P\)-polynomial with parameter \(t\) is obtained from the Macdonald \(P\)-polynomial by substituting the parameter \(q\) with \(0\).

import Math.Algebra.Hspray
import Math.Algebra.SymmetricPolynomials
n = 4
lambda = [2, 2]
macPoly = macdonaldPolynomial' n lambda 'P'
poly = changeParameters macPoly [zeroSpray, qlone 1]
hlPoly = hallLittlewoodPolynomial' n lambda 'P'
asSimpleParametricSpray poly == hlPoly
-- True

We use asSimpleParametricSpray because, contrary to the Hall-Littlewood \(P\)-polynomial, the Macdonald \(P\)-polynomial is not represented by a SimpleParametricSpray a spray but by a ParametricSpray a spray, because its coefficients are not polynomials in the two parameters \(q\) and \(t\), but ratios of polynomials.

Combinatorics

The modules Math.Combinatorics.Kostka and Math.Combinatorics.Tableaux appeared in version 1.4.6.0. They provide some functions to compute Kostka-Jack numbers, possibly skew, to enumerate the semistandard Young tableaux with a given shape and a given weight, possibly skew, and to enumerate Gelfand-Tsetlin patterns. The reason to include these modules in the package is that these functions are used to compute the symmetric polynomials.

References

• I.G. Macdonald. Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, second edition, 1995.

• J. Demmel and P. Koev. Accurate and efficient evaluation of Schur and Jack functions. Mathematics of computations, vol. 75, n. 253, 223-229, 2005.

• The symmetric functions catalog. https://www.symmetricfunctions.com/index.htm.