The alar is essentially a term for an arcanists mind when being applied to sympathy. Whenever a sympathetic link is created, a part of the alar must be "split" from the rest in order to believe that the linked objects are the same. Kvothe often describes himself as having an "alar like a blade of Ramston steel," or a "razor sharp alar," referring to the fact that he was able to split his alar into many pieces in order to simultaneously link multiple objects. This is useful in a variety of applications.
Kvothe's promotion to E'lir at the university is a result of his displaying his ability to split his alar two ways, which he does as a demonstration for Master Hemme' s introductory sympathy class. He now has the ability to split his alar five ways, and perhaps six.