= Solution
Use upper indices for the defining <representation> of <SU(6)> and lower indices for its <dual representation>. The <Young tableaux> below display tensor-slot labels; their shapes determine the <Young diagrams>. Full columns of height six are <determinant representations> and can be removed for <SU(6)>.
$$
\begin{array}{c|c|c|c|c}
\text{representation}&\text{Dynkin labels}&\text{tableau}&\text{tensor}&\dim\\\hline
\mathbf6&(1,0,0,0,0)&\boxed{1}&v^a&6\\[3pt]
\overline{\mathbf6}&(0,0,0,0,1)&\begin{array}{l}\boxed{1}\\[-3pt]\boxed{2}\\[-3pt]\boxed{3}\\[-3pt]\boxed{4}\\[-3pt]\boxed{5}\end{array}&w_a&6\\[3pt]
\mathbf{21}&(2,0,0,0,0)&\boxed{1}\boxed{2}&S^{(ab)}&21\\[3pt]
\mathbf{15}&(0,1,0,0,0)&\begin{array}{l}\boxed{1}\\[-3pt]\boxed{2}\end{array}&A^{[ab]}&15\\[3pt]
\mathbf{35}&(1,0,0,0,1)&\begin{array}{l}\boxed{1}\boxed{2}\\[-3pt]\boxed{3}\\[-3pt]\boxed{4}\\[-3pt]\boxed{5}\\[-3pt]\boxed{6}\end{array}&T^a{}_b,\ T^a{}_a=0&35
\end{array}
$$
Here all indices run from one to six. The <highest weight> <Dynkin labels> follow from the differences of successive row lengths. In the same order the row lengths are $(1)$, $(1,1,1,1,1)$, $(2)$, $(1,1)$ and $(2,1,1,1,1)$.
The defining <vector> transforms as $v^a\mapsto U^a{}_bv^b$. For a <unitary representation>, the <conjugate representation> is equivalent to the <dual representation>, so $\overline{\mathbf6}$ can be written as a <covector> with $w_a\mapsto w_b(U^{-1})^b{}_a$. Its five-box column is equivalently a totally <antisymmetric tensor> $A^{a_1\cdots a_5}$, related by the invariant alternating <tensor> $w_a=\epsilon_{aa_1\cdots a_5}A^{a_1\cdots a_5}/5!$.
The <symmetric tensor> transforms as $S^{ab}\mapsto U^a{}_cU^b{}_dS^{cd}$, with $6(6+1)/2=21$ independent components. The <antisymmetric tensor> has the same transformation law but obeys $A^{ab}=-A^{ba}$, giving $6(6-1)/2=15$ independent components. For the <Adjoint representation>, $T\mapsto UTU^{-1}$; the <trace> constraint removes one component from the $36$ entries of a mixed <tensor>, leaving $35$. Its six-box <Young diagram> is obtained by converting the five-index column to a lower index and selecting the traceless mixed <tensor>.
Thus the requested <dimensions>, in the order above, are
$$
\boxed{6,\quad6,\quad21,\quad15,\quad35.}
$$
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