Embeddings of root systems. I: Root systems over commutative rings.
The paper under review considers a class of reflection representations of Coxeter groups over commutative unital rings \(R\) and their corresponding root systems. For \(R=\mathbb{R}\), these include the standard (real, reduced) root systems. In general, they also include, for any (unital) subring \(Z\) of \(\mathbb{R}\), certain root systems over coefficient rings \(R\) which are generalized \(Z\)-fusion algebras arising as quotients of coproducts over \(Z\) of Verlinde algebras (over \(Z\)) for \(\text{su}(2)\) WZW conformal field theories at levels determined by entries of the Coxeter matrix, call these \(Z\)-fusion root systems and \(Z\)-fusion reflection representations. Any \(Z\)-fusion root system \(\Phi\) of a Coxeter system \((W,S)\) embeds \(Z\)-linearly in a standard real root system \(\Psi\) (defined over \(Z\)) of a corresponding Coxeter system \((W',S')\). Every Coxeter system \((W,S)\) has \(Z\)-fusion root systems for every subring \(Z\) of \(\mathbb{R}\). In particular, \((W,S)\) has a canonical \(\mathbb{Z}\)-fusion root system \(\Phi\) (its ``universal symmetric root system over commutative rings''); for this \(\Phi\), \((W',S')\) is a simply-laced Coxeter system. In addition, any Coxeter system for which all the entries of its Coxeter matrix are either 1, 2, 3, 4, 6, \(\infty\) or odd primes has a \(\mathbb{Z}\)-fusion root system \(\Phi\) whose coefficient ring \(R\) is a subring of \(\mathbb{R}\), and for which \(\Phi\) is just a standard root system. As an example, the paper describes explicitly an embedding \(\Phi(H_4)\to\Phi(E_8)\) of standard root systems.
scientific article; zbMATH DE number 5599086
scientific article