Jack Symmetric function and expansion in power symmetric basis
The question involves SageMath and its Symmetric functions package. Particularly I am interested in Jack symmetric functions. So to define Jack symmetric functions in SageMath:
sage: Sym = SymmetricFunctions(QQ)
sage: Zonal = Sym.jack(t=2).J()
sage: Schur = Sym.jack(t=2).J()
sage: s = Sym.schur()
My interest in all these functions as expansion in power symmetric basis hence
sage: P = Sym.p() # to expand symmetric functions in power symmetric basis
sage: P(Zonal([2,2]))
p[1, 1, 1, 1] + 2*p[2, 1, 1] + 7*p[2, 2] - 8*p[3, 1] - 2*p[4]
Similarly
sage: P(Schur([2,2]))
1/12*p[1, 1, 1, 1] + 1/4*p[2, 2] + (-1/3)*p[3, 1]
Now the problem is that p[2,1,1] is equal to $p_2 p_1^2$
but with the Sage representation I cannot work by replacing
$p_i$ with $p_i/h$ for example. As I know the character formula
for Schur functions, I wrote my own programme using symmetrica.charvalue
and for shur([2,1,1]) it returns
$$
1/8p_1^4/h^4 - 1/4p_1^2p_2/h^3 - 1/8p_2^2/h^2 + 1/4*p_4/h
$$
My question is: can I do it without writing a whole code just by using the predefined Schur function that is jack symmetric function for $t = 1$?
The second question is about the same thing for $t = 2$ Zonal polynomials.
The zonal polynomial can be written in a power symmetric basis in SageMath,
but I want it as a product of the $p_i$, not p[Partition]. Is there
a formula that exists in literature to do that or can I just do it
in SageMath using the existing Jack function?
I can rewrite the code if I have a reference for zonal polynomial in a power symmetric basis.
