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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}}}
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    $\begingroup$ You are basically asking about the full list of invariants and mixed concomitants of an alternating 3-form in 6 variables. I would try to look at the book by Gurevich on invariant theory as well as the more classical book by Turnbull. Also Rota and his school studied invariants of antisymmetric tensors using so called letter-place algebras. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Mar 1, 2018 at 18:07
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    $\begingroup$ $\operatorname{Sym}^7 \left(\wedge^3 \mathbb{C}^6\right)$ has the representation corresponding to partition $\left(5, 5, 5, 2, 2, 2\right)$ appearing twice. Thus, not very multiplicity-free. $\endgroup$
    – darij grinberg
    Commented Mar 1, 2018 at 19:49
  • $\begingroup$ There may be a link with Farey fractions. $\endgroup$
    – Sylvain JULIEN
    Commented Mar 1, 2018 at 20:41
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    $\begingroup$ try also alexandria.tue.nl/repository/freearticles/588258.pdf which indicates there are finitely many projective orbits. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Mar 1, 2018 at 23:49

3 Answers 3

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$$\bigoplus_{k \ge 0} t^k Sym^{k} V = \frac{1}{(1−tV)(1−t^2 \mathbf{g})(1−t^3 V)(1−t^4)(1−t^4 V_2)},$$ 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.

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    $\begingroup$ In what ring is this identity to be interpreted? It does not seem true in the representation ring, as the dimension of the left side is much smaller than the dimension of the right. $\endgroup$
    – Will Sawin
    Commented Mar 2, 2018 at 17:17
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    $\begingroup$ @WillSawin If I understand the paper of Landsberg and Manivel correctly, the RHS should be understood "mnemonically", using the rule $V_\lambda V_\mu=V_{\lambda+\mu}$ on irreducible modules. (I had the same concerns about dimensions, so I downloaded the paper and tried to figure out what was going on.) I suppose one can impose a filtration of the representation ring, and take the associated graded. Also, I think this formula is valid over $SL(6)$, not $GL(6)$. $\endgroup$
    – Vladimir Dotsenko
    Commented Mar 2, 2018 at 21:49
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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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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.

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