Characteristic multivectors of Coxeter transformations give novel insights into the geometry of root systems	

Pierre-Philippe Dechant	-- University of Leeds, Maths	

There has been increased recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of symmetry structures: the Coxeter transformations. We  calculate the invariants for the bipartite Coxeter transformations for $A_8$, $D_8$, $E_8$ and $A_6$, $D_6$, $E_6$. We focus on bivector invariants in particular, and shed new light on the relationships with other well-known invariant planes, including the Coxeter plane, as well as recent work into orthogonal decomposition. I will also briefly present results from our recent paper that calculated invariants of all Coxeter elements exhaustively and analysed the resulting computational algebra dataset using data science techniques such as Neural Networks and Principal Component Analysis.