# Computer algebra system for Tensor Algebra

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?