There's no such thing as "alpha", or whatever. We (human beings) aren't wolves, and the whole "alpha male" thing is just psuedoscientific nonsense.
What you perceive to be an alpha male, or omega, is really nothing more than simple self-confidence, which is independent of a person's social personality type (introvert or extrovert).
You can have a person who is an introvert and socially demure with plenty of self confidence.
Likewise you can have a person who is an extrovert and socially dominant, but lacks any significant self confidence.
The "alpha male" is a myth, made up to perpetuate the stereotype of the socially dominant extrovert, stolen from a failed study of wolf packs and nonsensically applied to human social interaction.
We are more complex than that, and as it turns out, so are the wolves they were studying.