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

TL;DR

This paper proves Feigin and Shoikhet's conjecture that the quotients of the lower central series of a free associative algebra exhibit polynomial growth, using representation theory and computational verification.

Contribution

It provides a proof of the polynomial growth conjecture and establishes bounds on the structure of irreducible modules in the series.

Findings

Confirmed polynomial growth of lower central series quotients.

Established bounds on the number of squares in Young diagrams for modules.

Validated the conjectured module structure for specific cases using computational methods.

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 $B_3(A_3)$ conjectured in math/0610410 .