# What breaks down in the theory of affine hyperplane arrangments?

It appears to me that there is a substantial amount of combinatorial algebra and geometry supporting the theory of central hyperplane arrangements (See Topics in Hyperplane Arrangements, Aguiar and Mahajan).

What breaks when extending this theory to the affine case? Is there a simple relationship between the affine and central case which makes all of the results of Tits algebras/incidence algebras/Dynkin idempotents apply?

I ask this as a machine learning theorist very interested in leveraging many of the algebraic and combinatoric results of this beautiful field, but in my domain we care strictly about the affine case. Since it appears many of the texts and papers in this area concern central arrangements, I'd like to know if those results somehow apply trivially to the affine case, before I embark on developing a deep understanding of the field.

(I'll also note that there is some relationship between the Tits semigroup of an affine arrangement and subsemigroup of the cone of the arrangment, but is this enough to use a large majority of the results in the field? eg. JKS isomorphism etc. My knowledge of the implications of the foregoing semigroup relationship is too limited to make a judgement on if I should precede.)

## 1 Answer

Good question! The main advantage of using a central arrangement seems to be that you have a nice unit (the center), getting a monoid instead of just a semigroup. I'm not sure when, if ever, that difference causes a theorem to fail later on.

In the book you mention by Aguiar and Mahajan, they show that even when the arrangement is not central, the linearization has a unit. So the Tits algebra for affine arrangements shares at least that property. Of course the book is the best source, but I drew some small examples in my review. To quote myself, "The unit is found as the formal linear combination of all the bounded faces, positive for the even dimensional faces and negative for the odd. (If the arrangement is not essential, we use rank instead of dimension.)" https://www.ams.org/journals/bull/0000-000-00/S0273-0979-2018-01660-1/S0273-0979-2018-01660-1.pdf