Root pairing

Summary

A root pairing is a family of roots in a module together with matching coroots in a dual module. Each root and its coroot define a reflection, and these reflections permute the roots and the coroots. It extends the notion of a root system from real Euclidean spaces to modules over any commutative ring, with no inner product. Over the integers, with free modules of finite rank and finitely many roots, it is the same as a root datum. Root data classify reductive algebraic groups.