Nov. 12, 2023. The Fourth International Conference on Groups, Graphs and Combinatorics/The Fourth International Conference on Group Actions and Symmetrical Graphs.
Southern University of Science and Technology. Shenzhen, Guangdong Province, China.
Abstract:
A representation of a Coxeter group is called a reflection representation if each of the defining generators acts by an abstract reflection. In this talk, we classify the isomorphism classes of reflection representations (over the real or complex number field) using the homology groups of certain graphs. Under some mild conditions, these representations correspond to Lusztig's $a$-function value 1, that is, correspond to the second-highest two-sided cell of the Coxeter group. If time permits, we also present some other results and problems about these representations, e.g., a generalization of Steinberg's theorem on their exterior powers.