What Is a Coxeter Group?

Coxeter groups arise in a wide variety of areas, so every mathematician should know some basic facts about them, including their connection to “Dynkin diagrams.” Proofs about these “groups generated by reflections” mainly use group theory, geometry, and combinatorics. This talk will briefly explain: what it means to say that G is a “group generated by reflections” (or, equivalently, that G is a “Coxeter group”); what it means to say that G is “of type An, Bn, or Dn”, and that all but finitely many of the “irreducible” finite groups generated by reflections are either dihedral groups or belong to these types. For further reading, see [1, Chap. 1], [2], or [3]. 1. What is a group generated by reflections? Example. In R2: The reflection across the y-axis is the map „x;y…, „ x;y…. The reflection across the line y ƒ x is the map „x;y…, „y;x…. In general, it is easy to see that the reflection L across a line L is the unique isometry of R2 that fixes each point on the line L, and has order 2 (i.e., 2 L ƒ Id, but L ” Id). Definition. Let H be a hyperplane in Rn. (E.g., a line in R2 or a plane in R3.) The corresponding reflection H is the unique isometry of Rn that fixes each point on H, and has order 2. Remark. If H passes through the origin, and e is a unit vector that is orthogonal to H, then H„x… ƒ x 2„x e… e: Example. Let L1 and L2 be two lines in R2, and assume ‹L1L2 ƒ =m, for some m 2 Z‚. Then (using i as a shorthand for Li) 1 2 is a rotation through angle 2 =m, so „ 1 2… ƒ Id. Indeed, h 1; 2i is the dihedral group D2m, so it has the presentation D2m ƒ h 1; 2 j 2 1 ƒ 2 2 ƒ „ 1 2… ƒ 1 i: This is a (fairly trivial) example of a finite group that is generated by reflections. In this theory, it is denoted I2„m…: the subscript 2 means that we are in R2, and them tells us that the order of 1 2 ism. Combined with the fact that reflections 1