For the moment we work over the complex numbers.
Suppose that $G = \mathrm{SL}(V)$, or $G = \mathrm{Sp}(V)$, or $G = \mathrm{SO}(V)$.
Weyl gave explicit constructions of irreducible representations of a given heighest weight in terms of the faithful representation $V$. This is done using Schur functors and possibly intersecting them with kernels of contractions maps. Roughly speaking, Weyl gives an *algorithm* to construct all irreducible representations, starting from the tautological representation.

My question is about a generalisation of such constructions to other base fields, and other `starting representations'. All representations are assumed to be algebraic. I am not exactly sure about what I am looking for. So I will give a suggestion for a definition, but it is rather clunky. Maybe there is already a crystal clear concept out there, that describes what I am looking for.

Let $K$ be a field of characteristic $0$. Let $G$ be a reductive group over $K$. Let $V$ be a representation of $G$ (defined over $K$), the `starting representation'. Now we can start building a category from $V$.
(Note, this a subcategory of $\mathrm{Rep}_{K}(G)$, but it is not by definition a *full* subcategory).

To this category $\mathscr{C}$ of `algorithmically constructible' representations we add:

- $V$
- tensor powers of objects in $\mathscr{C}$
- sums of objects in $\mathscr{C}$
- duals of objects in $\mathscr{C}$
- contraction maps $W^{\otimes n} \to W^{\otimes m}$, for $W \in \mathscr{C}$
- morphisms that come from the action of $\mathfrak{S}_{n}$ on $W^{\otimes n}$, for $W \in \mathscr{C}$
- kernels and cokernels of morphisms in $\mathscr{C}$

Maybe there are other natural algorithmic steps that should be added to this list, but I have not thought of any. These steps allow us to define Schur functors, so we find $\mathbb{S}^{\lambda}(V) \in \mathscr{C}$.

It is a well-known fact that if $V$ is faithful, then every representation of $G$ is a subobject of some object in $\mathscr{C}$, because $G$ is reductive. However, $\mathscr{C}$ is not closed under taking subobjects (in particular it need not be Tannakian).

I am not per se interested in knowing *which* algorithm would construct a given representation $W$, but rather in whether such an algorithm exists.
Thus, let us say that $V$ is an `algorithmically generating' representation if the inclusion functor $\mathscr{C} \hookrightarrow \mathrm{Rep}_{K}(G)$ is essentially surjective.

Now we get to my questions. In answering these questions, feel free to make additional assumptions on $G$. (E.g., semisimple, or absolutely simple.)

- Does every group admit an `algorithmically generating' representation?

My other question is more specific. Weyl looked at the classical groups, but (as far as I know) he omitted the spin groups. Over a general field, the spin representation need not be defined over $K$, but the (even) Clifford algebra is. Therefore:

- Let $(V,q)$ be a quadratic space (i.e., a $K$-vector space $V$ with a nondegenerate quadratic form $q$). Is the spin representation of $\mathrm{Spin}(V,q)$ on the even Clifford algebra $\mathrm{Cl}^{+}(V,q)$ an `algorithmically generating' representation?

Finally:

- Is there anything that I am forgetting in my definition of the category $\mathcal{C}$ of `algorithmically constructible' representations? Is there a known concept that I am overlooking?