nLab Coxeter transformation

Definition

If $W$ is a Coxeter group of finite rank then a Coxeter element, sometimes also called Coxeter transformation, is a product of all simple reflections in any order. Notice that this is not the same as the longest Coxeter element (the maximal element in the Bruhat order).

One talks about Coxeter transformations also in a number of more general situations. We say that a finite rank abelian group $A$ is a bilinear group if it is equipped with a (typically nonsymmetric) biadditive map

$$\langle -,-\rangle : A\times A\to\mathbf{Z}$$

and an automorphism $\tau:A\to A$, called the Coxeter transformation, satisfying

$$\langle y, x\rangle = -\langle x, \tau y\rangle,\,\,\,x,y\in A.$$

If $\langle -,-\rangle$ is nondegenerate then it follows that $A$ is torsion free hence a lattice, and we say that it is a bilinear lattice.

Examples

…talk about Euler bilinear form on the Grothendieck group of a suitable Abelian or triangulated category. In particular, about the Coxeter transformation induced by Serre duality.

…Auslander-Reiten translation induces Coxeter transformation…

Properties

If $A$ is a bilinear lattice then $\tau$ belongs to the center of $Aut(A)$.

Literature

• A. J. Coleman, Killing and the Coxeter transformation of Kac-Moody algebras, Invent. Math. 95, no. 3 (1989): 447–477
• N. A’Campo, Sur les valeurs propres de la transformation de Coxeter, Invent. Math. 33(1) (1976) 61–67 doi
• (around page 71) Claus Michael Ringel, Tame algebras and integral quadratic forms, Springer LNM 1099

Chapter VIII.2 in

• Maurice Auslander?, Idun Reiten?, Sverre O. Smalo, Representation theory of Artin algebras, Cambridge University Press 1995 (2010 online doi)
• Helmut Lenzing, A K-theoretic study of canonical algebras, RG, in book: Representation theory of algebras (Cocoyoc, 1994); CMS Conf. Proc. 18; Eds. R. Bautista, R. Martínez-Villa, J. A. de La Peña
• Andrzej Mróz, José Antonio de la Peña, Periodicity in bilinear lattices and the Coxeter formalism, doi
• Xinhong Chen, Ming Lu, Coxeter transformations of the derived categories of coherent sheaves, Journal of Algebra 399 (2014) 79-101 DOI
• Sefi Ladkani, On the periodicity of Coxeter transformations and the non-negativity of their Euler forms, pdf
• Michael Barot, Dirk Kussin, Helmut Lenzing, The Grothendieck group of a cluster category, arXiv:math.RT/0606518