jackpolynomials

Jack, zonal, Schur and skew Schur 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 and to compute them in symbolic form.

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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 (or parametric) Jack polynomial

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 $ prettyOneParameterQSpray "a" 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' $ evalOneParameterSpray jp 2
-- 18*x^3.y + 12*x^2.y^2 + 18*x.y^3

This is possible thanks to an upgrade of the hspray package which now provides the type OneParameterSpray (and more). An object of this type represents a multivariate polynomial whose coefficients depend on a parameter which is symbolically treated. The type of the Jack polynomial returned by the jackSymbolicPol function is OneParameterSpray a, and it is OneParameterQSpray for the jackSymbolicPol' function. The type OneParameterQSpray is an alias of OneParameterSpray Rational' where Rational' is a type defined in the numeric-prelude package, analogous to the well known Rational type.

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 OneParameterQSpray objects.

Note that if you use the function jackSymbolicPol to get a OneParameterSpray 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.Jack.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.Jack.SymmetricPolynomials
jp = jackSymbolicPol' 3 [3, 1, 1] 'J'
putStrLn $ prettySymmetricOneParameterQSpray "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.

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.