Lie powers and Witt vectors

Abstract

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 , . . .}, 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 kth tensor power V ⊗k .

A modular version of Klyachko's theorem on Lie representations of the general linear group

“…It is not obvious that the coefficients of this polynomial belong to Z, and it is not obvious that the coefficients are non-negative. Both of these facts were, however, proved in [2] by rather elaborate methods involving Witt vectors. Consequently B p m k (V ) is isomorphic to a direct sum of tensor products of the form…”

“…The decomposition is obtained by the use of Lyndon words. One of the advantages of this new approach to the main result of [2] is that it gives additional information about which products (…”

Module structure on Lie powers and natural coalgebra-split sub-Hopf algebras of tensor algebras

“…is an isomorphism. (2). There is a one-to-one correspondence, multiplicity preserving between summands of the functor L n and k(GL(V ))-summands of L n (V ).…”

“…There is a canonical connection between coalgebra decompositions of T and the decompositions of the Lie powers L n (V ) as modules over the general linear groups by restricting decomposition (1.1) to the primitives. The decompositions of Lie powers have been actively studied in the recent development of representation theory [2,3,4,9,10].…”

“…(1). The functor T H : V → T (V ) from modules to Hopf algebras; (2). The functor T C : V → T (V ) from modules to coalgebras by forgetting the multiplication; (3).…”