A polynomial criterion for Jordan elements in a free associative algebra
Abstract
Let J be the Jordan subalgebra of the free associative algebra Q⟨ X⟩ generated by X and $1$ under a· b=1/2(ab+ba). We give a polynomial criterion for an associative polynomial to belong to J. For every n≥1, Robbins' recursion gives a family of operators Tα∈Q[Sn], indexed by compositions of n into parts $1$ and $2$. We prove that right multiplication by Un=∑αTα† Tα has image SJ(n), the space of multilinear Jordan elements. If its characteristic polynomial is tenqn(t), where qn(0)≠0, then Πn=I−qn(Un)/qn(0) is an idempotent and f∈SJ(n) if and only if fqn(Un)=0, or equivalently fΠn=f. The same operators give the criterion for arbitrary homogeneous polynomials of degree n. We also prove SJ(n)≅⨁λ⊢ nrλSλ, where rλ=rankρλ(Un). Thus dimSJ(n)=∑λdλrλ and en=∑λdλ(dλ−rλ), with dλ=dim Sλ. 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.
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BibTeX
@misc{mashurov2026polynomial,
title = {{A polynomial criterion for Jordan elements in a free associative algebra}},
author = {F. Mashurov and B. Sartayev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.08314},
eprint = {2609.08314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08314},
}
RIS
TY - UNPB
TI - A polynomial criterion for Jordan elements in a free associative algebra
AU - F. Mashurov
AU - B. Sartayev
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2609.08314
UR - https://arxiv.org/abs/2609.08314
AB - Let J be the Jordan subalgebra of the free associative algebra Q⟨ X⟩ generated by X and $1$ under a· b=1/2(ab+ba). We give a polynomial criterion for an associative polynomial to belong to J. For every n≥1, Robbins' recursion gives a family of operators T_α∈Q[S_n], indexed by compositions of n into parts $1$ and $2$. We prove that right multiplication by U_n=∑_αT_α^† T_α has image SJ(n), the space of multilinear Jordan elements. If its characteristic polynomial is t^e_nq_n(t), where q_n(0)≠0, then Π_n=I−q_n(U_n)/q_n(0) is an idempotent and f∈SJ(n) if and only if fq_n(U_n)=0, or equivalently fΠ_n=f. The same operators give the criterion for arbitrary homogeneous polynomials of degree n. We also prove SJ(n)≅⨁_λ⊢ nr_λS^λ, where r_λ=rankρ_λ(U_n). Thus dimSJ(n)=∑_λd_λr_λ and e_n=∑_λd_λ(d_λ−r_λ), with d_λ=dim S^λ. 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.
ER -