LIE POWERS AND LIE MODULES
KAI MENG TAN
Let n, r ∈ Z+ , and let V be an n-dimensional vector space over an infinite field F
of prime characteristic p. The Lie power Lr (V ) is the homogeneous part of degree r
of the free Lie algebra L(V ) generated by V . This is a natural module for the Schur
algebra S(n, r). For n ≥ r, the Schur functor fr sends Lr (V ) to the Lie module
Lie(r) for the symmetric group Sr .
Our motivation of studying these modules come from the work of Selick and Wu.
They reduce the problem of finding natural homotopy decompositions of the loop
suspension of a p-torsion suspension to an algebraic question, and in this context
it is important to know a maximal projective submodule of Lie(n) when the field
has characteristic p. The Lie module also occurs naturally as homology of configuration spaces, and in other contexts. Moreover the representation theory of
symmetric groups over prime characteristic is difficult and many basic questions are
open; naturally occurring representations are therefore of interest and may give new
understanding.
The talk will present some of the recent results about these modules.
Theorem 1 (Bryant-Lim-T.). For r not a p-power, there is a direct summand
B r (V ) of Lr (V ) which is a direct sum of indecomposable tilting modules labelled by
p-regular partitions, and
dim(B r (V ))
→1
dim(Lr (V ))
as r → ∞ through the positive integers which are not p-powers.
Theorem 2 (Lim-T.). Let M be a S(n, r)-module. Let s ∈ Z+ . The Schur functor
frs sends the S(n, rs)-module Ls (M ) to
⊗s
rs
⊗F Lie(s)).
IndS
Sr oSs (fr (M )
Theorem 3 (Bryant-Lim-T.). For r not a p-power, there is a projective submodule
Br of Lie(r) such that
dim(B r )
→1
dim(Lie(r))
as r → ∞ through the positive integers which are not p-powers.
Theorem 4 (Erdmann-Lim-T.). Let Cr denote the complexity of Lie(r). Then
(1) Cpk ≤ k,
(2) Cpk m = max(Cp , . . . , Cpk ), where m ≥ 2 and p - m.
In particular, Cpk m ≤ k for all m ∈ Z+ .
2000 Mathematics Subject Classification. 20C30.
1
2
KAI MENG TAN
References
[1] R.M. Bryant, K.J. Lim, K.M. Tan, ‘Asymptotic behaviour of Lie powers and Lie modules’,
arXiv:1009.0974, submitted.
[2] K. Erdmann, K.M. Tan, ‘The non-projective part of the Lie module for the symmetric group’,
submitted.
[3] P. Selick, J. Wu, ‘On natural coalgebra decompositions of tensor algebras and loop suspensions’,
Mem. Amer. Math. Soc. 148 (2000), no. 701.
Department of Mathematics, National University of Singapore, Block S17, 10 Lower
Kent Ridge Road, Singapore 119076.
E-mail address: [email protected]