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. It can also compute their 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 $ prettySymbolicQSpray "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' $ evalSymbolicSpray jp 2
-- 18*x^3.y + 12*x^2.y^2 + 18*x.y^3

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. The type of these polynomials is SymbolicSpray, defined in the hspray package (which will be possibly renamed to ParametricSpray in the future).

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 $ prettySymmetricSymbolicQSpray "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.