I am trying to understand Young tableaux in the context of $SU(2)$ irreducible representations (this is a physics course so we use them just as a tool without much care in the meaning).
As I understand a Young tableau is an arrangement of boxes where each box represents an index of a tensor. Columns denote the antisymmetrization of the tensor indices, while rows the symmetrization of the indices.
To calculate the dimension of a particular representation we count all the possible standard tableaux associated to a particular form.
"In a standard tableaux indices do not decrease from left to right in a row, and always increase from top to bottom in a column. The tensors corresponding to nonstandard tableaux either vanish or are not independent of the standard tableaux, i.e. can be expressed in terms of these."source
What I don't understand is that if $i=1, j=3, k=3$, in the following diagram, the tensor is zero applying the symmetry propertie. Yet this is considered a standard tableaux (well in seems that in mathematical context they are called semistandard Young tableux, but the naming convention shouldn't be relevant).