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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.)

  1. 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:

  1. 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:

  1. 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?
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The short answers to your questions are all No.

The reason is that Weyl's construction works when the centraliser algebra of a tensor power of $V$ is a quotient of a Brauer algebra. This is similar to Schur's construction of representations of $SL(n)$ which assumes that the centraliser algebra is a quotient of the group algebra of the symmetric group.

In general, if $V$ is self-dual, then there is a homomorphism from the Brauer algebra to the centraliser algebra. The constructions you describe are equivalent to taking images of (primitive) idempotents in the Brauer algebra. For other representations the homomorphism is not surjective and there will be images which are reducible.

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  • $\begingroup$ Thanks for your answer! It is very instructive to know what the conceptual background is for Weyl's construction. $\endgroup$
    – user105976
    Mar 16 '17 at 8:01

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