The KP–BSO compatibility conjecture for multilinear identities

Let $\mathbb{F}$ be a coefficient field, let $n\geq 2$, and let $\omega$ be a multilinear $n$-ary operation in an associative algebra over $\mathbb{F}$. Fix $d\equiv 1\pmod{n-1}$, and for $1\leq e\leq d$ let $I_e$ be the space of multilinear degree-$e$ identities of $\omega$, let $\operatorname{KP}(I_e)$ be the result of applying the KP algorithm, and set

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

Applying the BSO algorithm to $\omega$ gives multilinear $n$-ary operations $\omega_1,\dots,\omega_n$ in an associative dialgebra; let $J_e$ be their multilinear degree-$e$ identities and set

$$J_d(\omega_1,\dots,\omega_n)=\bigoplus_{1\leq e\leq d}J_e.$$

KP–BSO compatibility conjecture. If the field $\mathbb{F}$ has characteristic $0$ or $p>d$, then

$$\operatorname{KP}_d(\omega)=J_d(\omega_1,\dots,\omega_n).$$

Equivalently, when the group algebra $\mathbb{F}S_d$ is semisimple, finding the identities of $\omega$ and applying the KP algorithm gives the same result as applying the BSO algorithm and finding the identities of the resulting operations. The conjecture asserts compatibility of these two procedures for multilinear identities of $n$-ary operations.