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https://github.com/stla/jackpolynomials

Jack, zonal, and Schur polynomials
https://github.com/stla/jackpolynomials

haskell jack-polynomials schur-polynomials zonal-polynomials

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Jack, zonal, and Schur polynomials

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# jackpolynomials



***Jack, zonal, Schur, and other symmetric polynomials.***



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Schur polynomials have applications in combinatorics and zonal polynomials have

applications in multivariate statistics. They are particular cases of

[Jack polynomials](https://en.wikipedia.org/wiki/Jack_function), 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`:



```haskell

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:



```haskell

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:



```haskell

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:



```haskell

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:



```haskell

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.



```haskell

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:

```haskell

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`.



```haskell

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`. 



```haskell

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.



```haskell

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 $0$.



```haskell

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.



### Kostka-Jack numbers, Kostka-Foulkes polynomials, and Kostka-Macdonald polynomials



The package can also compute the Kostka-Jack numbers (i.e. Kostka numbers with

a Jack parameter), the Kostka-Foulkes polynomials (aka $t$-Kostka polynomials), 

the Kostka-Macdonald polynomials (aka $qt$-Kostka polynomials), and the 

Hall polynomials. Skew generalizations are also available: skew Kostka-Jack 

numbers, skew Kostka-Foulkes polynomials, and skew Kostka-Macdonald polynomials.



### Semistandard Young tableaux and Gelfand-Tsetlin patterns 



The package allows to enumerate the semistandard Young tableaux with a given 

shape, possibly skew, and a given weight, and to enumerate Gelfand-Tsetlin 

patterns defined by a skew partition.



## 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. .