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### Jack Symmetric function and expansion in power symmetric basis

The question is involving using Sagemath and its Symmetric function package. Particularly I am interested in Jack symmetric function. So to define Jack symmetric function in sagemath

~~~

Sym = SymmetricFunctions(QQ)

Zonal = Sym.jack(t=2).J()

Schur = Sym.jack(t=2).J()

s = Sym.schur()

~~~

My interest in all these functions as expansion in power symmetric basis hence

~~~

P= Sym.p() #P() will expand the symmetric function in power symmetric basis.

~~~ $P(Zonal([2,2]))$ will give output $$p[1, 1, 1, 1] + 2p[2, 1, 1] + 7p[2, 2] - 8p[3, 1] - 2p$$ Similarly $P(Schur([2,2]))$ $$1/12p[1, 1, 1, 1] + 1/4p[2, 2] + (-1/3)*p[3, 1]$$

Now the problem is that $p[2,1,1]$ it's equal to $p_2 p_1^2$ but with the sage representation I can not work by replacing $p_i$ with $p_i /h$ for example. As I know the character formula for Schur funciton I wrote my own programme using ${\bf{symmetrica.charvalue}}$ and it returns for $shur([2,1,1])$

$$1/8p_1^4/h^4 - 1/4p_1^2p_2/h^3 - 1/8p_2^2/h^2 + 1/4*p_4/h$$

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 I can 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. 2 None

### Jack Symmetric function and expansion in power symmetric basis

The question is involving using Sagemath involves SageMath and its Symmetric function package. functions package. Particularly I am interested in Jack symmetric function. functions. So to define Jack symmetric function in sagemath

~~~

functions in SageMath:

sage: Sym = SymmetricFunctions(QQ) SymmetricFunctions(QQ)
sage: Zonal = Sym.jack(t=2).J() Sym.jack(t=2).J()
sage: Schur = Sym.jack(t=2).J() Sym.jack(t=2).J()
sage: s = Sym.schur() ~~~Sym.schur()


My interest in all these functions as expansion in power symmetric basis hence

~~~

P=

sage: P = Sym.p() #P() will  # to expand the symmetric function in power symmetric basis. ~~~
$P(Zonal([2,2]))$ will give output
$$p[1, symmetric functions in power symmetric basis sage: P(Zonal([2,2])) p[1, 1, 1, 1] + 2p[2, 2*p[2, 1, 1] + 7p[2, 7*p[2, 2] - 8p[3, 8*p[3, 1] - 2p$$
Similarly
$P(Schur([2,2]))$
$$1/12p[1, 2*p  Similarly sage: P(Schur([2,2])) 1/12*p[1, 1, 1, 1] + 1/4p[2, 1/4*p[2, 2] + (-1/3)*p[3, 1]$$1]


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