I've just started learning about Lie theory (only just finished up to basic classification of semisimple lie algebras) and I've got the following questions:
How do I show that the complex lie algebra of type $E_6$ has 78 dimensions?
What is the fundamental representation of $E_6$ group and why does it has 27 dimensions?
I understand that these questions might seems a bit annoying to answer so if there's no simple answer could someone recommend me any readings for further lie theory that will focus on exceptional lie algebras/groups? (preferably a maths textbook/lecture note but with some application to physics as well because that's ultimately what I will be doing).