On an n-ary generalization of the Lie representation and tree Specht modules
Year: 2026
Type: article
Abstract: We continue our study, initiated in our prior work with Richard Stanley, of the representation of the symmetric group on the multilinear component of an n -ary generalization of the free Lie algebra known as the free Filippov n -algebra with k brackets. Our ultimate aim is to determine the multiplicities of the irreducible representations in this representation. This had been done for the ordinary Lie representation ( n = 2 case) by Kraskiewicz and Weyman. The k = 2 case was handled in our prior work, where the representation was shown to be isomorphic to S 2 n − 1 1 . In this paper, for general n and k , we obtain decomposition results that enable us to determine the multiplicities in the k = 3 and k = 4 cases. In particular we prove that in the k = 3 case, the representation is isomorphic to S 3 n − 1 1 ⊕ S 3 n − 2 21 2 . Our main result shows that the multiplicities stabilize in a certain sense when n exceeds k . As an important tool in proving this, we present two types of generalizations of the notion of Specht module that involve trees.
Source: Advances in Mathematics
Authors: Tamar Friedmann, Phil Hanlon, Michelle L. Wachs
Institutions: Colby College, Dartmouth College, University of Miami
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,790
Funder: National Science Foundation
OpenAlex ID: W4393026743