I am reading Deligne & Milne's notes on tannakian categories. In this document, the authors defin the notion of Deligne categories, $\underline{\mathrm{Rep}}(GL_t)$, let me give a rough introduction to this sort of categories and what I am confused about.
One of the main interest of this paper is neutral Tannakian categories, which can be defined in two equivalent ways:
- A category $\mathcal{C}$ which is rigid abelian tensor such that $\mathrm{End}(\mathbf{1})=k$ is a field ($\mathbf{1}$ is "the" identity object) and $\mathcal{C}$ is equipped with an exact faithful $k$-linear tensor functor $\mathcal{C} \to \mathrm{Vect}_k$, the latter category indicates the category of finite dimensional vector spaces over $k$. Such a functor, if exists, is called a fiber functor.
- The category $\mathrm{Rep}_k(G)$ consists of finite dimensional representations of a given affine group scheme $G$ over $k$, i.e. its objects are pairs $(V,\rho)$, $\mathrm{dim}_k(V) < +\infty$ and $\rho: G \to GL_V$ is considered as a morphism of functors $k-\mathbf{Alg} \to \mathbf{Grp}$. The fiber functor in this case is just the forgetful functor.
The purpose of the last example $1.27$ in chapter $1$ in DM's note is to show a counterexample, i.e. a rigid tensor abelian category such that $\mathrm{End}(\mathbf{1})$ is a field but it can not be endowed a fibered functor. The key point is to look at categorical ranks of objects. Ranks in $\mathrm{Rep}_k(G)$ are nonnegative integers, here Deligne creates catgories with ranks are intermediates.
Let denote $\mathcal{C}_0$ be the category with objects are symbols $T^{a,b}$ ($a,b \in \mathbb{N}$), morphism sets $$\mathrm{Hom}(T^{a,b},T^{c,d}) = \begin{cases} \mathbb{Z}[t][S_{a+d}] & \text{if} \ a + d = b + c \\ 0 & \text{other cases} \end{cases}.$$ The tensor product is simply $T^{a,b} \otimes T^{c,d} = T^{a+c,b+d}$. Define $\mathcal{C} = \mathcal{C_0}^{add}$ be the additive envelope of $\mathcal{C_0}$, i.e. formally adding direct sums. This category fails the positivity because $\mathrm{rank}(T^{1,0})=t$ is not in $\mathbb{N}$, the ground ring, however, is $\mathbb{Z}[t]$, not a field so we extend it to $\mathbb{Q}(t)$ to obtain a new category, let denote it by $\mathcal{C} \otimes \mathbb{Q}(t)$. Finally, we consider the Karoubian envelope $(\mathcal{C} \otimes \mathbb{Q}(t))^{kar}=\underline{\mathrm{Rep}}(GL_t)$ (formally adjoining images of idempotents) this category has several properties:
- $\mathrm{rank}(T^{1,0})$ is not an integer, of course!
- The endormophism set of every object is semisimple algebra by using Maschke theorem, as they are isomorphic to $\mathbb{Q}(t)[S]$ for certain symmetric group $S$.
- It is Karoubian, i.e. every idempotent has image.
First question. That is, this category is "almost" rigid abelian tensor and what I want to prove is to prove it is indeed abelian, I don't have a clue to this so many thanks for any suggestion.
Second question. Deligne wrote down an isomorphism for $$\mathrm{Hom}(T^{a,b},T^{c,d}) \otimes_{\mathbb{Z}[t]} \mathbb{C} \to \mathrm{Hom}_{GL_V}(V^{\otimes a} \otimes V^{* \otimes b}, V^{\otimes c} \otimes V^{* \otimes d}).$$ in which $V$ is finite-dim $\mathbb{C}$-vector space and $n=\mathrm{dim}(V) \geq a + d$, $\mathbb{C}$ is a $\mathbb{Z}[t]$ algebra under the morphism $\mathbb{Z}[t] \to \mathbb{C}, t \mapsto n$. My question here is what exactly is the morphism above? Why should we consider the action of $GL_V$ here? I guess the RHS is somehow related to Schur-Weyl duality but I cannot write it explicitly. I don't even think we need this isomorphism to conclude $\mathrm{End}(\bigoplus T^{a,b}) \otimes_{\mathbb{Z}[t]} \mathbb{C}$ is semisimple for some sum $\bigoplus T^{a,b}$.
Update. The first question is done, it is a special of theorem $2$ in U. Jannsen, Motives, numerical equivalence, and semi-simplicity.
If $\mathcal{C}$ is $k$-linear, Karoubian and endomorphism semisimple then $\mathcal{C}$ is abelian semisimple.
