Let $G$ be a split reductive group over an arbitrary field $k$. By definition, see Jantzen (\*), an ascending chain $$0 = V\_0 \subset V\_1 \subset V\_2 \subset \dots$$ of submodules of a $G$-module $V$ is called a Weyl filtration of $V$ if $V = \bigcup\_{i \geq 0} V\_i$ and if each $V\_i/V\_{i-1}$ is isomorphic to some Weyl module $V(\lambda\_i)$ of highest weight $\lambda\_i$.


It seems that if a module admits such a Weyl filtration, we have $V \cong \bigoplus\_{i} V(\lambda\_i)$. Indeed, according to remark II.4.19 in Jantzen, we can order the $\lambda\_i$ such that $\lambda\_i < \lambda\_j$ implies $i > j$. The decomposition $V \cong \bigoplus\_{i} V(\lambda\_i)$ then follows by induction from $\operatorname{Ext}^1\_G(V(\lambda),V)=0$ if $V$ has no weight $\mu > \lambda$ (see Jantzen II.2.14 Remark 2).


In particular I would like to use that we can write the symmetric square of a Weyl module as a direct sum of Weyl modules and that we can derive the terms from Weyl's character formula. This should follow from Proposition II.4.21 in Jantzen which basically says (the dual of) if $V$ admits a Weyl filtration then so does $V \otimes V$.


However, this results seems quite standard and I am quite surprised that Jantzen doesn't mention it (as far as I know). So my question is whether there is a more direct approach towards this problem and whether anyone knows a good reference.


EDIT. The reasoning in the second paragraph is not correct. We would want the $\lambda\_i$ to be ordered such that if $\lambda\_i < \lambda\_j$ then $i < j$. I am not sure whether this is possible. Probably it is not, which explains why this isn't in Jantzen.


(\*) Jantzen, Representations of Algebraic Groups (AMS, 2003)



Decomposition into Weyl modules
Your conclusion about direct sums is false. It's helpful here to have some examples in mind, such as a typical projective/injective module for the Lie algebra of $G:=\mathrm{SL}\_2$, lifted to $G$. This is *indecomposable* but might have a Weyl filtration with two quotients: for example, the trivial 1-dimensional module $V(0) =L(0)$ at the top, but the linked Weyl module $V(2p-2)$ at the bottom. (Here I am referring to a non-negative integral multiple of the single fundamental weight just by the integer coefficient, while $L$ as in Jantzen denotes a simple module.)


In any case, you seem to be referring to the expanded second edition of Jantzen's 1987 Academic Press book *Representations of Algebraic Groups* (AMS, 2003), where the results you want are in Part II. For example, he refers to 4.19 as II.4.19 if the part is unclear from the context. 


Note too that Jantzen's treatment is for a connected *semisimple* algebraic group, though adaptations to a reductive group with nontrivial (semisimple) derived subgroup can be deduced easily. And the field of definition doesn't play a role here.


