2
$\begingroup$

I have been studying hyperplane arrangements from R. Stanley's notes and so far, I have read until lecture 5. But, Lecture 6 and end of Lecture 5 seems very combinatorial. I'm more interested in the "algebraic" direction. More specifically, in representation-theoretic perspective (if there exists any).

So, I was wondering if someone can suggest me some reference for more "algebraic" perspective of hyperplane arrangements. (It's OK if it's not representation-theoretic)

Improve this question
$\endgroup$
1
  • 3
    $\begingroup$ Orlik and Terao's Arrangements of Hyperplanes? $\endgroup$
    – user61318
    Commented Feb 24, 2022 at 8:27

1 Answer 1

Reset to default
2
$\begingroup$

If I recall correctly, Stanley does not discuss the so-called "Tits monoid" associated to a hyperplane arrangement. The relationship between the Tits monoid and the lattice of flats of a hyperplane arrangement is the main focus of the book "Topics in hyperplane arrangements" by Aguiar and Mahajan (it's free online at http://pi.math.cornell.edu/~maguiar/surv-226.pdf). Actually, they have another more recent book on hyperplane arrangements as well - I'm not quite sure the difference between the two. Anyways, this is definitely an "algebraic" direction in hyperplane arrangement theory.

The study of free arrangements/the module of logarithmic derivations -- as championed by Terao -- is another completely different algebraic (or more specifically, algebro-geometric) direction in hypreplane arrangement theory. This one is arguably more connected (at least "morally") to the stuff in Stanley's notes (e.g. I believe Stanley mentions free arrangements, at least in an exercise). This is discussed e.g. in the book by Orlik and Terao mentioned in the comment above.

Improve this answer
$\endgroup$
3
  • 1
    $\begingroup$ By the way, a perhaps superficial distinction between these two directions: first one concerns arrangements over $\mathbb{R}$, second one concerns arrangements over $\mathbb{C}$. $\endgroup$
    – Sam Hopkins
    Commented Feb 24, 2022 at 17:01
  • 1
    $\begingroup$ The Tits monoid essentially views a hyperplane arrangement as an oriented matroid. An oriented matroid is defined as a monoid of covectors with some extra properties. I don't know if there is any reasonable algebraic way to detect if an oriented matroid is realizable from a hyperplane arrangement. These monoids come up in our monogragph ams.org/books/memo/1345/memo1345.pdf (and there is a semigroup for affine arrangements). $\endgroup$
    – Benjamin Steinberg
    Commented Mar 2, 2022 at 19:14
  • 1
    $\begingroup$ @BenjaminSteinberg: your link requires a sign-in. But I believe the book in question is "Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry" by Margolis, Saliola, and Steinberg (mathscinet.ams.org/mathscinet-getitem?mr=4365944). $\endgroup$
    – Sam Hopkins
    Commented May 4, 2022 at 14:23

You must log in to answer this question.