Let $z_1, z_2, \cdots z_N$ be $N$ variables obeying the condition $z_i^M=1$ for some positive integer $M>N$. Let $F_N$ be the space of antisymmetric polynomials of these variables. Given a set $E = (e_1,e_2,\cdots e_N) \subset (0,1,\cdots M-1) $ of size $N$, we define $f_E=\det z_i^{e_j} \in F_N$. Using this basis $\{f_E\}$ of $F_N$ we define a map
$ \varphi : F_N \rightarrow F_{M-N} \ : \ \varphi(f_E) = s_E f_{E^c} $
where $E^c$ is the complement of $E$ in $(0,1,\cdots M-1) $, and $s_E$ is some nonzero $z_i$-independent factor.
Is the following conjecture true, and does it have a simple proof?
Conjecture: There exist factors $s_E$ such that $\varphi$ is a morphism of three-algebras, namely $ \forall f,g,h \in F_N, \qquad \varphi(fgh) = \varphi(f)\varphi(g)\varphi(h) $
Motivations and remarks
- The conjecture can be tested in particular cases. I did this in the cases $(M,N)=(4,2), (5,2), (6,2), (6,3), (7,2), (7,3), (8,3), (8,4), (9,4)$.
- The morphism $\varphi$ can not be extended to a morphism of two-algebras acting on spaces of non necessarily antisymmetric functions. This can already be seen in the example $(M,N)=(4,2)$.
- The motivation for this problem is the study of Laughlin wavefunctions, which are relevant for the fractional quantum Hall effect. The idea of using the variables $z_i$ is due to M. Dyakonov.
