From the Young tableaux to the algebra

One can compute the the weights by filling in the numbers 1,2,3 according to the rule for semi-standard tableaux (not decreasing along the rows, strictly increasing down the columns). Each of the eight possible tableaux gives the eigenvalues of $lambda_3$ (the number of 1's minus the number of 2's) and $lambda_8$ (number of 1's plus number of 2's minus twice the number of 3's all divided by $sqrt 3$). If you plot them you will recognise the weight diagram of the octet (adjoint) rep.