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Abstract

Parallel to the very rich theory of Kazhdan-Lusztig cells in characteristic $0$, we try to build a similar theory in positive haracteristic. We study cells with respect to the $p$-canonical basis of the Hecke algebra of a
crystallographic Coxeter system (see arXiv:1510.01556(2)). Our main technical
tool are the star-operations introduced by Kazhdan-Lusztig which have interesting numerical consequences for the p$-canonical basis. As an application, we explicitely describe $p$-cells in finite type $A$ (i.e. for symmetric groups) using the Robinson-Schensted correspondence. Moreover, we show that Kazhdan-Lusztig cells in finite types $B$ and $C$ decompose into $p$-cells for $p > 2$.