Covering points in permutation algebras

Isomorphisms of fusion subcategories on permutation algebras

“…There is also a primitive idempotent (Q) , where w 1 has defect group P, while t 1 has defect group Q. The point containing w 1 covers Br Q (c) and lifts to α, a point of G on A that covers c. This is a consequence of [3,Theorem 3.5]. Notice that α verifies the axioms of Definition 5.1.…”

“…Remark 5.2. If i ∈ A G covers j than the entire (A G ) * -conjugacy class of i covers j as in [3,Definition 3.2]. Consequently a point α ⊆ A G covers j, if α contains an idempotent that verifies a) and b) in Definition 5.1.…”

“…Section 4 proves the first main result of the paper, that is Theorem 4.7. Sections 5 and 6 gather information on covering points on permutation algebras, that were introduced in [3]. Finally, in Section 7 we prove the second main result of the paper, namely Theorem 7.3.…”

Remarks on the extended Brauer quotient

“…We adopt here the definition of covering points from [1]. We say that the point α of G on A covers β if α has defect group P and for any i ∈ α there is an idempotent j 1 ∈ A N that lies in the conjugacy class of β and there is a primitive idempotent f ∈ A N belonging to a point with defect group P such that j 1 f = f j 1 = f and i f = f i = f .…”

“…Since N is normal in G, hence N N (P) is also normal in N G (P), the fact that this bijection restricts to a bijection between the points that cover β andβ is an easy verification given by [1,Theorem 3.5].…”