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,
HallLittlewood polynomials, and Macdonald
polynomials. In addition, it provides some functions to compute KostkaJack
numbers, KostkaFoulkes polynomials, KostkaMacdonald polynomials, Hall
polynomials, and to enumerate GelfandTsetlin 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 nonevaluated 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 ]
HallLittlewood polynomials
The package can also compute the HallLittlewood polynomials. A HallLittlewood
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 HallLittlewood
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 HallLittlewood \(P\)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 HallLittlewood \(P\)polynomial: the HallLittlewood \(P\)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 HallLittlewood
\(P\)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 KostkaJack
numbers, possibly skew, to enumerate the semistandard Young tableaux with a
given shape and a given weight, possibly skew, and to enumerate
GelfandTsetlin 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, 223229, 2005.

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