Errata to " Hyperplane Arrangements and Descent Algebras "

Hyperplane arrangements and descent algebras Franco V Saliola saliola-DesAlgLectureNotes.pdf version of 10 January 2006 Errata and addenda by Darij Grinberg I will refer to the results appearing in the article " A Hyperplane arrangements and descent algebras " by the numbers under which they appear in this article. 6. Errata • Various places (for example, §2.1): You use the notations ⊆ and ⊂ synonymously. It might be better if you consistently keep to one of them, as the appearance of both of them in your notes suggests that ⊂ means proper inclusion (but it does not). • Page 3: Replace " the nonempty intersections of the open half spaces " by " a nonempty intersection of open half spaces ". (Maybe also add " (one for each hyperplane) " at the end of the sentence.) • Page 4, Figure 3: I think the " (+0−) " label is wrong, and should be a " (−0−) " label instead. • Page 4: Replace " and that the closure " by " and that the closures ". • Page 5, §1.3: Your claim that " the join X ∨ Y of X and Y is X + Y " is generally false (even when A is the braid arrangement) 1. I don't think the join can be characterized this easily. (Of course, the existence of a join follows from the existence of the meet using the fact that any finite meet-semilattice having a greatest element is a lattice.) • Page 5, §1.3: I don't think your claim that " The rank of X ∈ L is the dimension of the subspace X ⊂ R d " is true. • Page 8, Exercise 2: I think it would be useful to add the following claim between (2) and (3): " x ≤ xy ". • Page 9: In the formula for σ H ij (BC), why do you write " C (j) < C (i) " instead of " C (i) > C (j) " ? Of course, this is equivalent, but it looks out of place.