jackpolynomials

Jack, zonal, and Schur polynomials with Python.

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 get these polynomials.

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The Jack polynomial in three variables of the integer partition $(2,1)$, with parameter $3/2$:

>>> from gmpy2 import mpq
>>> from jackpy.jack import JackPol
>>>
>>> poly = JackPol(3, [2, 1], alpha = mpq(3, 2))
>>> print(poly)
Poly(7/2*x_1**2*x_2 + 7/2*x_1**2*x_3 + 7/2*x_1*x_2**2 + 6*x_1*x_2*x_3 
 + 7/2*x_1*x_3**2 + 7/2*x_2**2*x_3 + 7/2*x_2*x_3**2, x_1, x_2, x_3, domain='QQ')

By default, JackPol returns the Jack $J$-polynomial. It can also return the Jack $C$-polynomial, the Jack $P$-polynomial, and the Jack $Q$-polynomial, by using the which argument.

The expression of a Jack polynomial can be long. Since a Jack polynomial is symmetric, it is possible to write it as a linear combination of some monomial symmetric polynomials, and this gives a shorter expression.

>>> from jackpy.jack import JackPol
>>> from jackpy.monomial_symmetric_polynomials import msp_combination_expr
>>> poly = JackPol(4, [2, 2], alpha = 5)
>>> print(poly)
Poly(84*x_1**2*x_2**2 + 28*x_1**2*x_2*x_3 + 28*x_1**2*x_2*x_4
 + 84*x_1**2*x_3**2 + 28*x_1**2*x_3*x_4 + 84*x_1**2*x_4**2
 + 28*x_1*x_2**2*x_3 + 28*x_1*x_2**2*x_4 + 28*x_1*x_2*x_3**2
 + 24*x_1*x_2*x_3*x_4 + 28*x_1*x_2*x_4**2 + 28*x_1*x_3**2*x_4 
 + 28*x_1*x_3*x_4**2 + 84*x_2**2*x_3**2 + 28*x_2**2*x_3*x_4 
 + 84*x_2**2*x_4**2 + 28*x_2*x_3**2*x_4 + 28*x_2*x_3*x_4**2 
 + 84*x_3**2*x_4**2, x_1, x_2, x_3, x_4, domain='QQ')
>>> expr = msp_combination_expr(poly)
>>> print(expr)
24*M[1;1;1;1] + 28*M[2;1;1] + 84*M[2;2]

Here the symbol M[1;1;1;1] represents the monomial symmetric polynomial of the integer partition $(1,1,1,1)$.

As of the development version 0.1.0.9000, it is possible to get Jack polynomials with a symbolic Jack parameter. These polynomials have domain 'QQ(alpha)', where alpha is the Jack parameter.

>>> from jackpy.jack import JackSymbolicPol
>>>
>>> poly = JackSymbolicPol(3, [2, 2])
>>> print(poly)
Poly((2*alpha**2 + 6*alpha + 4)*x_1**2*x_2**2 
 + (4*alpha + 8)*x_1**2*x_2*x_3 
 + (2*alpha**2 + 6*alpha + 4)*x_1**2*x_3**2 
 + (4*alpha + 8)*x_1*x_2**2*x_3 
 + (4*alpha + 8)*x_1*x_2*x_3**2 
 + (2*alpha**2 + 6*alpha + 4)*x_2**2*x_3**2, 
 x_1, x_2, x_3, domain='QQ(alpha)')