Modular Lie powers and the Solomon descent algebra

Abstract

Let V be an r -dimensional vector space over an infinite field F of prime characteristic p, and let L n (V ) denote the nth homogeneous component of the free Lie algebra on V . We study the structure of L n (V ) as a module for the general linear group G L r (F) when n = pk and k is not divisible by p and where r ≥ n. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of L k (V ) and the indecomposable direct summands of L n (V ) which are not isomorphic to direct summands of V ⊗n . Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups.

The Decomposition of Lie Powers

“…By the same proof as for [14,Theorem 3], there is a surjective homomorphism of algebras c r,R : D r,R → C r,R satisfying c r,R (X ν ) = φ ν,R for all compositions ν of r.…”

“…For definitions and further details see [14, §2] and the works cited there. As well as results from [14] we shall use a theorem of Garsia and Reutenauer [15]. Our notation is similar to that in [14, §2].…”

The module structure of the Solomon–Tits algebra of the symmetric group

“…Note that, in this case, P ∧ Q = Q I 1 ∨ · · · ∨ Q I l . For example, if A = {3, 5, 6, 7, 8}, Q = (8, 67, 3, 5) and P = (367, 58), then Q P , I = (23,14) and P ∧ Q = (67, 3,8,5).…”

“…However, the little we know has recently proved to be of tremendous help in the study of modular Lie representations of arbitrary groups [12,14]. It would therefore be desirable to obtain more information on the modular Solomon algebra.…”

“…A huge body of research papers on the subject, accumulated during the past fifteen years, provides surprising links to many different fields in geometry, combinatorics, algebra and topology ( [2,3,10,14,15,[17][18][19][20]23], to name but a few; see [22,26] for a more exhaustive list of references).…”