A polynomial criterion for Jordan elements in a free associative algebra

Abstract

Let $J$ be the Jordan subalgebra of the free associative algebra $\Q\langle X\rangle$ generated by $X$ and $1$ under $a\cdot b=\frac{1}{2}(ab+ba)$. We give a polynomial criterion for an associative polynomial to belong to $J$. For every $n\ge 1$, Robbins' recursion gives a family of operators $T_α\in\Q[S_n]$, indexed by compositions of $n$ into parts $1$ and $2$. We prove that right multiplication by $U_n={\sum }_{\alpha }{T}_{\alpha }^{†}{T}_{\alpha }$ has image $\SJ(n)$, the space of multilinear Jordan elements. If its characteristic polynomial is $t^{e_n}{q}_n(t)$, where $q_n(0)\ne 0$, then ${\Pi }_n=I-q_n(U_n)/q_n(0)$ is an idempotent and $f\in\SJ(n)$ if and only if $f{q}_n(U_n)=0$, or equivalently $f{\Pi }_n=f$. The same operators give the criterion for arbitrary homogeneous polynomials of degree $n$. We also prove $\SJ(n)\cong\bigoplus_{λ\vdash n}r_λS^λ$, where $r_λ=\rankρ_λ(U_n)$. Thus $\dim\SJ(n)=\sum_λd_λr_λ$ and $e_n={\sum }_{\lambda }{d}_{\lambda }(d_{\lambda }-r_{\lambda })$, with $d_{\lambda }=\dim S^{\lambda }$. We give a recursion for the corresponding row spaces and use it to compute the multiplicities over $\Q$ through degree $12$. Exact block characteristic polynomials through degree $7$ and direct computations in small degrees illustrate the criterion.

arXiv commentRevised version with new results. We welcome comments, suggestions, and discussion