Lie powers and Witt vectors

Bryant, R.M. ; Johnson, Marianne

Journal of Algebraic Combinatorics, Tome 28 (2008) no. 1, pp. 169-187.

Résumé

Summary: In the study of Lie powers of a module $V$ in prime characteristic $p$, a basic role is played by certain modules $B _{ n }$ introduced by Bryant and Schocker. The isomorphism types of the $B _{ n }$ are not fully understood, but these modules fall into infinite families ${ B _{ k}, B _{ pk}, B _{ p $^2$ k},\dots }$ {B_k,B_pk,B_p^2k,$\dots $} , one family $B( k)$ for each positive integer $k$ not divisible by $p$, and there is a recursive formula for the modules within $B( k)$. Here we use combinatorial methods and Witt vectors to show that each module in $B( k)$ is isomorphic to a direct sum of tensor products of direct summands of the $k$th tensor power $V ^{ \otimes k }$.

Détail

Keywords: keywords free Lie algebra, Lie power, tensor power, Witt vector