Recently, Xie Xun, Associate Researcher of the School of Mathematics and Statistics, BIT, published a research paper entitled Conjectures P1-P15 for Coxeter Groups with Complete Graph in the international authoritative academic journal Advances in Mathematics, in which, it is proved that Lusztig’s conjectures P1-P15 holds for Coxeter groups with complete graph.

Based on some known results of isoparametric Hecke algebras, Lusztig proposed a series of conjectures about the KL basis and a-function of non-isoparametric Hecke algebras in the 1990s, which were later called Conjectures P1-P15. The Conjectures P1-P15 in the case of being isoparametric can be derived from the positive conjecture that was solved by Elias and Williamson in 2014. It is known that the positive conjecture is only valid in the case of being isoparametric, therefore, the Conjectures P1-P15 in the case of being non-isoparametric is still an important open issue.

In this paper, Xie Xun established a method to prove Conjectures P1-P15 for Coxeter groups with complete graph, whose unique feature is the descending induction of a-functions. Several newly discovered lemmas on a-functions play a vital role in the calculation of a-functions of Coxeter groups with complete graph, which which is of great significance to understand the a-functions of general Coxeter groups. Besides, a simple characterization of the left, right and bilateral cavities of Coxeter groups with complete graph is displayed, which is the main progress in this research.

This research is funded by the Academic Start-up Program of BIT and the Youth Fund of the National Natural Science Foundation of China.

Paper link:https://doi.org/10.1016/j.aim.2021.107565

Profile:

Xie Xun, Associate Researcher, member of Algebra Team of the School of Mathematics and Statistics, BIT, has long been engaged in the research of algebraic groups, quantum groups, and Hecke algebra, and has made breakthroughs in many special cases of Lusztig’s Conjectures P1-P15. Currently, under the support of the Youth Fund of the National Natural Science Foundation of China, he has published several papers in Advances in Mathematics, International Mathematics Research Notices, Journal of Algebra, and Journal of Pure and Applied Algebra.