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}}}

```


What is known about the plethysm $\text{Sym}^d(\bigwedge^3 \mathbb{C}^6)$
$$\bigoplus\_{k \ge 0} t^k Sym^{k} V = \frac{1}{(1−tV)(1−t^2 \mathbf{g})(1−t^3 V)(1−t^4)(1−t^4 V\_2)},$$
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](https://projecteuclid.org/euclid.mmj/1091112085).


