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# Root systems, affine subspaces, and projections

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### Abstract

We tackle several problems related to a finite irreducible crystallographic root system $$\Phi$$ in the real vector space $$\mathbb E$$. In particular, we study the combinatorial structure of the subsets of $$\Phi$$ cut by affine subspaces of $$\mathbb E$$ and their projections. As byproducts, we obtain easy algebraic combinatorial proofs of refinements of Oshima's Lemma and of a result by Kostant, a partial result towards the resolution of a problem by Hopkins and Postnikov, and new enumerative results on root systems.

### Author and article information

###### Journal
02 September 2021
###### Article
10.1016/j.jalgebra.2021.07.035
2109.00944
cc206f17-8c3b-4f94-aeac-a557fa612c8c