# jackpolynomials

*Jack, zonal, Schur and skew Schur 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). This package
allows to evaluate these polynomials. It can also compute their symbolic form.

___

Evaluation of the Jack polynomial with 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 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.


### Symbolic (or parametric) Jack polynomial

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 $ 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:

```haskell
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:

```haskell
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>.