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Using symmetric function package and multiplication in sagemath

asked 2022-02-13 04:12:01 +0200

Anupamsage gravatar image

updated 2023-05-19 14:29:29 +0200

FrédéricC gravatar image

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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answered 2022-02-13 16:24:38 +0200

Max Alekseyev gravatar image

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)
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Thanks a lot

Anupamsage gravatar imageAnupamsage ( 2022-02-18 01:25:00 +0200 )edit
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answered 2022-02-13 08:50:38 +0200

rburing gravatar image

Define

Sym = SymmetricFunctions(SR)

instead.

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Thanks a lot

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

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Asked: 2022-02-13 04:12:01 +0200

Seen: 199 times

Last updated: Feb 13 '22