<s>[INST] What is known about the plethysm $\text{Sym}^d(\bigwedge^3 \mathbb{C}^6)$ What is known about the plethysm $\text{Sym}^d(\bigwedge^3 \mathbb{C}^6)$ as a representation of $\text{GL}(6)$? It is my understanding that this should be multiplicity-free. I tried computing it using the Schur Rings package in Macaulay2 and I cannot see a pattern among the weights that appear.
If a formula is known, a reference would be nice also. Thanks.
EDIT: To save others the work, here is the data for $0 \leq d \leq 5$:
{{0, 0, 0, 0, 0, 0}}, 
{{0, 0, 0, 1, 1, 1}}, 
{{0, 1, 1, 1, 1, 2}, {0, 0, 0, 2, 2, 2}}, 
{{1, 1, 1, 2, 2, 2}, {0, 1, 1, 2, 2, 3}, {0, 0, 0, 3, 3, 3}}, 
{{2, 2, 2, 2, 2, 2}, {1, 1, 2, 2, 3, 3}, {1, 1, 1, 3, 3, 3}, {0, 2, 2, 2, 2, 4}, {0, 1, 1, 3, 3, 4}, {0, 0, 0, 4, 4, 4}}, 
{{2, 2, 2, 3, 3, 3}, {1, 2, 2, 3, 3, 4}, {1, 1, 2, 3, 4, 4}, {1, 1, 1, 4, 4, 4}, {0, 2, 2, 3, 3, 5}, {0, 1, 1, 4, 4, 5}, {0, 0, 0, 5, 5, 5}}}

 [/INST] No, it is not multiplicity-free. Already for $d=6$, this representation contains the Schur functor $S^{4,4,4,2,2,2}$ twice. This can be easily checked in Magma (even the online calculator) issuing the commands 
Q := Rationals();
s := SFASchur(Q);
s.[6]~s.[1,1,1];
Ignoring the Schur functors that vanish on $\mathbb{C}^6$, we obtain
s.[4,3,3,3,3,2] + 2*s.[4,4,4,2,2,2] + s.[5,4,3,3,2,1] + s.[5,4,4,2,2,1] +
s.[5,5,4,2,1,1] + s.[5,5,5,1,1,1] +s.[6,3,3,3,3] + s.[6,4,4,2,2] + s.[6,5,5,1,1] +s.[6,6,6]
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