If each quadra had a "representative enneagram type" what would it be?
The way I see it is that alpha has a need for outside approval, and generally wants to brighten the mood of those nearby. Even the intellectual alphas ultimately want to be recognized and liked for their accomplishments, no matter how esoteric, and are happy to share their knowledge.
6w5 seems to embody the anxiety a lot of betas have with abuse of authority, dishonesty, loyalty, and a general "us vs. them" mentality
Gamma is similar to alpha in that they ultimately want recognition, but it's for more concrete and work related achievements. Alphas might just want to be liked, but gamma is all about accomplishment and esteem.
delta: 9w1 (or 1w9?)
Deltas want everyone to get along and to live their quiet lives in peace. There really isn't anything especially contentious about delta, but they take their ideals seriously and will work for them.