jackpolynomials

Jack, zonal, Schur, skew Schur, and Hall-Littlewood polynomials.

Schur polynomials have applications in combinatorics and zonal polynomials have applications in multivariate statistics. They are particular cases of Jack polynomials. This package allows to evaluate these polynomials as well as the Hall-Littlewood polynomials and to compute them in symbolic form. It also provides some utilities for symmetric polynomials.

Evaluation of the Jack polynomial with 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 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.

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, with a symbolic Jack polynomial:

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 parameter. It is known that the Jack polynomials with Jack parameter \(\alpha\) are orthogonal for the Hall inner product with 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 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' :: SimpleParametricQSpray
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

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.

• Jack polynomials. https://www.symmetricfunctions.com/jack.htm.