I was wondering if anyone knows where I can find a formula for the dimension of an irreducible module of highest weight $\Lambda$ expressed only in terms of the Young diagram corresponding to $\Lambda$.


Thank you. 



Dimension of $\mathfrak{sl}_n$ modules
One influential older source to consult is the concise Springer Lecture Notes No. 682 by Gordon James *The Representation Theory of the Symmetric Groups* (1978). Section 26 applies the symmetric group theory to general linear groups, using the language of "Weyl modules" and partitions. See in particular his Theorem 26.5 for the dimension formula in terms of semistandard partitions. It takes some work, however, to get into his notational scheme, though his overall approach to the subject has the virtue of leading naturally into fields of prime characteristic while also treating the classical Schur-Weyl duality.


Note by the way that in the classical setting there is no essential difference between the group and Lie algebra theories here, while the transition from general linear to special linear is quite easy.


ADDED: I'm still unsure what is really needed, but I don't see the partition formulation of weights for general (or special) linear groups or Lie algebras to be helpful in considering duality. In the notes by James, or in many textbooks like Fulton-Harris (with varying notation), one studies the irreducible representations of a *fixed* $GL\_d$ on the $n$-th tensor powers of the natural $d$-dimensional module for variable $n$ by using the representation theory of the symmetric groups $S\_n$. Here a partition of $n$ into at most $d$ parts corresponds nicely to a highest weight. But it's tricky to bring the transpose (conjugate) partition into this picture, since its number of parts might exceed $d$.


The role of the transpose partition for $S\_n$ itself is more straightforward: Partitions of $n$ naturally index irreducible representations of $S\_n$. Then the transpose partition indexes the product of the given representation and the sign representation, with no change in dimension. But the relationship of this machinery (via Schur functors) to representations of each fixed $GL\_d$ is indirect. 


