= Solution
Use the standard chain ordering of the five simple <roots> of $SU(6)$. A <Young diagram> of row lengths $r_1\geq\cdots\geq r_6=0$ has <highest weight> <Dynkin labels> $a_i=r_i-r_{i+1}$. A full column of six boxes can be removed because it is the trivial <determinant representation>. The required data are
$$
\boxed{\begin{array}{c|c|c|c|r}
\text{representation}&(a_1,a_2,a_3,a_4,a_5)&\text{row lengths}&\text{tensor}&\dim\\\hline
\mathbf6&(1,0,0,0,0)&[1]&v^i&6\\
\bar{\mathbf6}&(0,0,0,0,1)&[1,1,1,1,1]&w_i&6\\
\mathbf{21}&(2,0,0,0,0)&[2]&S^{ij}=S^{ji}&21\\
\mathbf{15}&(0,1,0,0,0)&[1,1]&A^{ij}=-A^{ji}&15\\
\mathbf{35}&(1,0,0,0,1)&[2,1,1,1,1]&M^i{}_j,\ M^i{}_i=0&35
\end{array}.}
$$
In the same order, the <Young tableaux> are
$$
\square,\qquad
\begin{array}{l}\square\\[-4pt]\square\\[-4pt]\square\\[-4pt]\square\\[-4pt]\square\end{array},\qquad
\square\,\square,\qquad
\begin{array}{l}\square\\[-4pt]\square\end{array},\qquad
\begin{array}{l}\square\,\square\\[-4pt]\square\\[-4pt]\square\\[-4pt]\square\\[-4pt]\square\end{array}.
$$
Each upper index transforms by $U$, and each lower index by $U^{-1}$. In particular $M\mapsto UMU^{-1}$ is the complex <Adjoint representation>. The column of five boxes can be represented either by $w_i$ or by the totally <antisymmetric tensor> $B^{i_1\ldots i_5}$, using the invariant six-index alternating tensor:
$$
w_j=\frac1{5!}\epsilon_{j i_1\ldots i_5}B^{i_1\ldots i_5}.
$$
Thus it really is the <conjugate representation>, rather than a second independent six-dimensional choice. Similarly converting the lower index of $M^i{}_j$ into five alternating upper indices gives the displayed hook <Young diagram>; the removed <trace> would be its full six-box column component.
The dimension counts are
$$
\dim\operatorname{Sym}^2\mathbb C^6=\frac{6\cdot7}{2}=21,\qquad
\dim\Lambda^2\mathbb C^6=\frac{6\cdot5}{2}=15,\qquad
\dim\operatorname{End}_0(\mathbb C^6)=6^2-1=35.
$$
The first two count independent entries of <symmetric tensors> and <antisymmetric tensors>; the last removes one <trace> from a mixed <tensor>. The <hook-content formula> independently gives these dimensions from the <Young diagrams>.
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