# Using symmetric function package and multiplication in sagemath

I am having a problem working in SageMath with the following series. In Maple, I could define

$$n(d):= \sum_{k=1}^{d} \operatorname{schur}((k,1^{d-k}))\prod_{\square \in (k,1^{d-k})} \, G(c(\square)h) \tag{*}$$

where

• $\operatorname{schur}$ is the usual schur function I have created using the character formula,
• $G(h)$ is a series in $h$,
• the $c(\square)$ is the content of the Young tableaux.

For example for $d=3$ it can be read as follows

$$\operatorname{schur}([1,1,1]) \, G(h) \, G(2h) - \operatorname{schur}([2,1]) \, G(h) \, G(-h) + \operatorname{schur}([3]) \, G(2h) \, G(h)$$

I wanted to do a similar thing in SageMath with Jack polynomials instead of Schur polynomials.

Hence I defined

Sym = SymmetricFunctions(QQ)
JJ = Sym.jack(t=1).J()
s = Sym.schur()


I can see that I have defined the symmetric function over the field of rationals. Even when I define a new variable in Sage with var('h') I cannot multiply JJ[2,1] and h -- it gives an error -- hence I cannot compute $(*)$ with Jack polynomials in SageMath.

I am sure there is a way around. Please let me know.

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I'd advise against using symbolic variables. Since you want power series in h, you can define a ring of them upfront and then symmetric functions over that ring:

R.<h> = PowerSeriesRing(QQ)
Sym = SymmetricFunctions(R)

more

Thanks a lot

( 2022-02-18 01:25:00 +0200 )edit

Define

Sym = SymmetricFunctions(SR)


more

Thanks a lot

( 2022-02-18 01:24:53 +0200 )edit