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}}}

 A: 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]
A: A trivial remark: for $d=4k$ the module $[2k,2k,2k,2k,2k,2k]$ should appear. This is because of the degree four relative invariant listed in Proposition 7 page 81 of the classic and very long paper by Sato and Kimura on Prehomogeneous vector spaces.
The quartic invariant is described explicitly in Ring of invariants of $\operatorname{SL}_6$ acting on $\Lambda^3 \mathbb C^6$ 
Robert Bryant mentions in that question that the quartic invariant generates the invariant ring. This should imply that the only modules $[m,m,m,m,m,m]$ you will see for general $d$ are the ones I just described, with multiplicity one.
A: $$\bigoplus_{k \ge 0} t^k Sym^{k} V = \frac{1}{(1−tV)(1−t^2 \mathbf{g})(1−t^3 V)(1−t^4)(1−t^4 V_2)},$$
where $V = \wedge^3 \mathbb{C}^6 = [0,0,1,0,0], V_2 = [0,1,0,1,0]$, and $\mathbf{g} = [1,0,0,0,1].$
See section 6 of "Series of Lie Groups" by Landsberg and Manivel.
