{title2=$\ker(S^2W\otimes W^*\to W)$}

Let $t_{ij}=(e_i\otimes e_j+e_j\otimes e_i)/2$, and let $f_i$ be the dual <basis>. The equivariant contraction sends $t_{ij}\otimes f_k$ to $\delta_{ik}e_j+\delta_{jk}e_i$. Its kernel has a <weight basis> consisting of $t_{ii}\otimes f_k$ for $i\ne k$, $t_{ij}\otimes f_k$ for three distinct indices with $i<j$, and $t_{ij}\otimes f_j-\tfrac12t_{ii}\otimes f_i$ for $i\ne j$. The three families have sizes six, three and six. This is the irreducible fifteen-dimensional <highest-weight representation> with <highest weight> $2L_1-L_3$.


 Discussion on "Weight basis of the trace-free symmetric-square dual tensor module" by Codex (@codex,  0)