Computing plethysms of the adjoint representation using the Littlewood Richardson rule Let $N$ be an integer (let's imagine very large), and let $G$ be the group $\mathrm{GL}_N(\mathbb{C})$. I would like to compute various plethysms of irreducible representations which are not polynomial. As a typical example, if $M$ is the adjoint representation of $G$ (or the trace zero part of the adjoint), then I might be interested in decomposing $M \otimes M \otimes M$ into irreducible representations. The answer should be expressible in terms of partitions independent of $N$ if we assume that $N$ is large enough and we allow partitions with negative entries. Unfortunately, the sage Littlewood-Richardson calculator doesn't allow me to multiply $(1,-1)$ and $(1,-1)$ (say).
Question 1: Is there a trick to compute plethysms of non-polynomial representations on a computer using some implementation of the LR-rule? For example, one could multiply $(2,1,1,\ldots,1,0)$ by itself and then [somehow] judiciously extract the answer?
Question 2: This is in some sense a second question, but is also something that confuses me. Even for polynomial representations $V$, is it easy using LR to (numerically) compute $\wedge^n V$ for small $n$? (In practice, I would like to combine Question 1 and Question 2 to compute e.g. $(\wedge^3 M) \otimes M \otimes M$ for the adjoint representation. 
 A: If you only want to take tensor products of non-algebraic representations here is an answer:
Non-polynomial algebraic representations always become polynomial by tensoring with a large enough power of the determinant representation $D = (1,1,...,1)$. Tensoring with a one dimensional representation always preserves irreducibility, and in this case combinatorially corresponds to adding one to every entry of the partition. Similarly, tensoring with the inverse determinant corresponds to subtracting 1 from every part of the partition.
In the case of the adjoint representation $M$, $M \otimes D = D \oplus (2,1,1...,1,0)$. And for any $V$ we have that $M \otimes V = M \otimes D \otimes V \otimes D^{-1}$ (tensoring representations is associative and commutative up to isomorphism) so we can compute this in terms of the regular $LR$ rule for $(M \otimes D) \otimes V$ and then subtracting $1$ from each part of the corresponding partitions.
Computing more general plethysms is hard to do numerically, even in the case of polynomial representations.  
