Young tableaux of $8\otimes 8$ in $SU(3)$

Question

In Georgi's Lie Algebras in Particle Physics, one finds the following Young tableaux for $8\otimes 8$ in $SU(3)$:

I am unsure of all the cancellations. Let us number the canceled tableaus increasing from left to right and top to bottom. There are seven cancellations. I understand cancellations 4, 6 and 7 because they are antisymmetric fourth rank, which must vanish. For instance, I don't know why 2 gets cancelled but the one next to it, which has virtually the same structure, doesn't get cancelled.

Any help would be greatly appreciated.

Answer 1

I don't have a copy of Georgi to hand so I can not tell you his conventions. However there is a rule that for each cell the number of $b$'s above and to the right must be less than the number of $a$'s. The second cancellation you mention fails that. (It may be that Georgi uses a similar but equivalent rule.)

See for example page 12 bullet point 3 for exactly the same statement but worded differently. As I say, Georgi may have yet another!

Answer 2

Rule 2 in "Young-Tableau Methods for Kronecker Products of Representations of the Classical Groups" (FERMILAB-Pub-80/49-THY June 1980) says that when adding more than one box...

Label the boxes in the top row “a,” the next row “b,” etc. Add each box one by one, always in a one-box-permissible way, the top row first, then the 2nd row, etc., and such that reading from right to left and then up to down, the number of “a” boxes encountered is always $\geq$ the number of “b”‘s $\geq$ number of “c’s , and so on. (Emphasis mine.)

This rule explains cancellations 1, 2, 3, and 5.

I have not read Georgi's book, but since in the late 70's and early 80's he was doing Young Tableaux for SU(N) in the same way as I was, I assume he has a rule in his book that is very similar to my rule 2.