Bremner–Felipe–Sánchez-Ortega conjecture relating the KP and BSO algorithms

Let $\mathbb{F}$ be a field, let $\omega$ be a multilinear $n$-ary operation over $\mathbb{F}$, and fix a degree $d$. For $1\leq e\leq d$, let $A_e$ be the multilinear degree-$e$ subspace of the free nonassociative $n$-ary algebra on $e$ generators, and let $I_e\subseteq A_e$ consist of the polynomial identities satisfied by $\omega$. Let $B_e$ be the multilinear degree-$e$ subspace of the free nonassociative algebra with $n$ operations of arity $n$, and let $\mathrm{KP}(I_e)\subseteq B_e$ be obtained by applying the KP algorithm. Define

$$\mathrm{KP}_d(\omega)=\bigoplus_{1\leq e\leq d}\mathrm{KP}(I_e).$$

Applying the BSO algorithm to $\omega$ gives multilinear $n$-ary operations $\widehat{\omega}_1,\dots,\widehat{\omega}_n$. Let $J_e\subseteq B_e$ be the polynomial identities satisfied by these operations, and define

$$J_d(\widehat{\omega}_1,\dots,\widehat{\omega}_n)=\bigoplus_{1\leq e\leq d}J_e.$$

Bremner–Felipe–Sánchez-Ortega conjecture. If $\mathbb{F}$ has characteristic $0$ or $p>d$, then

$$\mathrm{KP}_d(\omega)=J_d(\widehat{\omega}_1,\dots,\widehat{\omega}_n).$$

The conjecture asserts that applying the KP algorithm to the identities of $\omega$ and directly computing the identities of the operations produced by the BSO algorithm give the same result. The characteristic assumption ensures the relevant group algebra $\mathbb{F}S_d$ is semisimple; the conjecture is presented as an open problem concerning the compatibility of these two constructions.