I'm looking at the notation given for Young tableau in Hubsch's book "Calabi-Yau Manifolds: A Bestiary for Physicists" - see Chapter 3, p. 96.

A Young tableau (for a $U(n)$ representation) is denoted by

$(b_1, \dots, b_n) \, , \quad b_r \leq b_{r+1} $

and the $b_{r}$ denote the number of boxes in the $r$-th row of the Young tableau, counting upwards from the bottom row.

This is all very clear until the mention of having negative values of $b_r$ - I've not really come across this before.

Further, the covariant ($v_\mu$) and contravariant ($v^\mu$) vector representations are denoted by

$(-1, 0, \dots , 0)$ and $(0, \dots, 0, 1)$


The only way this makes sense is if we consider "skew"-tableau, but I'm not too clear about how these map to the usual notion of representations. I'm sure this is well understood somewhere as I've seen this employed in calculations of cohomologies on projective spaces but I can't find a simpler explanation of the notation.


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