Two-Term Polynomial Identities
Allan Berele, Peter Danchev, Bridget Eileen Tenner
TL;DR
This work analyzes algebras satisfying two-term multilinear identities $x_1\cdots x_n = x_{\sigma(1)}\cdots x_{\sigma(n)}$, introducing eventual commutativity as a finitary strong form of commutativity. The authors develop a permutation-action framework, with $H_k$ capturing which monomials are identified by identities and the $T_i$ operators generating degree-raising consequences, to derive sharp bounds on the degree of eventual commutativity. For $q=1$ and $\sigma(1)\neq1,\sigma(n)\neq n$, they prove eventual commutativity of degree at most $2n-3$ (proved sharp via explicit constructions); for certain natural specializations (generalized commutativity and specific permutations) they obtain precise degrees like $n+1$ or $n+2$. When $q\neq 1$ the algebras are nilpotent, and the authors also confirm the Specht conjecture in the setting of eventual commutativity by proving finite generation of the corresponding $T$-ideals. These results connect two-term identities, permutation-structure in identities, and foundational consequences in PI-theory across characteristics.
Abstract
We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{σ(1)} \cdots x_{σ(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $σ$ not fixing 1 or $n$ are eventually commutative in the sense that the equality $x_1\cdots x_k = x_{τ(1)} \cdots x_{τ(k)}$ holds for $k$ large enough and all permutations $τ\in S_k$. Calling the minimal such $k$ the degree of eventual commutativity, we prove that $k$ is never more than $2n-3$, and that this bound is sharp. For various natural examples, we prove that $k$ can be taken to be $n+1$ or $n+2$. In the case when $q \ne 1$, we establish that the algebra must be nilpotent. We, moreover, demonstrate that if an algebra is eventually commutative of arbitrary characteristic, then it has a finite basis of its polynomial identities, thus confirming the Specht conjecture in this particular case.