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I am searching for a computer algebra system that allows to tackle the following problem:

Let $V = \mathbb{R}^3$. Consider the free commutative algebra $\mathrm{S}^\bullet V$, with $\mathrm{S}^kV$ being the $k$-fold symmetric tensor power of $V$. Now construct the tensor algebra $\mathrm{T}^\bullet(\mathrm{S}V)$, s.t. every $\phi \in \mathrm{T}^{(k_1,\dotsc,k_n)}(\mathrm{S}V)$ is of the form $$\phi = \phi_1 \otimes \dots \otimes \phi_n$$ with $\phi_i \in \mathrm{S}^{k_i}V$. A linear map $\Psi \colon \mathrm{S}^\bullet V \to W$ can be specified by defining it on generators of $\mathrm{S}^\bullet V$, and it can be extended to $\mathrm{T}^\bullet(\mathrm{S}V)$ as a (graded) derivation.

I now would like to compute the dimensions of the kernels of certain $\Psi$s which are defined by specifying them on generators as mentioned above. Is there a good computer algebra system to tackle that problem?

In the end $\mathrm{T}^{(k_1,\dotsc,k_n)}(\mathrm{S}V)$ is just a finite dimensional vector space and choosing a basis means I simply need to solve a linear system of equations. But is there a more abstract/symbolic way to treat this tensor products?

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