An upper bound for the lower central series quotients of a free associative algebra

2008/01/14 by Galyna Dobrovolska, G. Dobrovolska, Pavel Etingof +3 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT

paper · pdf · doi:10.48550/arxiv.0801.1997

7 pages; introduction expanded

openalex publication_date 2008/01/14 · arxiv created 2008/03/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Feigin and Shoikhet conjectured in math/0610410 that successive quotients B_m(A_n) of the lower central series filtration of a free associative algebra A_n have polynomial growth. In this paper we give a proof of this conjecture, using the structure of W_n-representation on B_m(A_n) which was defined in math/0610410 . We also prove that the number of squares in a Young diagram D corresponding to an irreducible W_n-module in the Jordan-Holder series of B_m(A_n) is bounded above by the integer (m-1)²+2[(n-2)/2](m-1). This bound combined with MAGMA computations by Rains in math/0610410 allows us to confirm the W_n-module structure of B₃(A₃) conjectured in math/0610410 .

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