Antisymmetric functions of the roots of unity: an elementary conjecture - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:00:14Z http://mathoverflow.net/feeds/question/55007 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55007/antisymmetric-functions-of-the-roots-of-unity-an-elementary-conjecture Antisymmetric functions of the roots of unity: an elementary conjecture Sylvain Ribault 2011-02-10T10:10:48Z 2011-02-10T10:10:48Z <p>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 </p> <p>$\varphi : F_N \rightarrow F_{M-N} \ : \<br> \varphi(f_E) = s_E f_{E^c}$</p> <p>where $E^c$ is the complement of $E$ in $(0,1,\cdots M-1)$, and $s_E$ is some nonzero $z_i$-independent factor.</p> <blockquote> <p>Is the following conjecture true, and does it have a simple proof? </p> </blockquote> <p><strong>Conjecture:</strong> There exist factors $s_E$ such that $\varphi$ is a morphism of three-algebras, namely $\forall f,g,h \in F_N, \qquad \varphi(fgh) = \varphi(f)\varphi(g)\varphi(h)$</p> <hr> <h2>Motivations and remarks</h2> <ol> <li>The conjecture can be tested in particular cases. I did this in the cases $(M,N)=(4,2), (5,2), (6,2), (6,3), (7,2), (7,3), (8,3), (8,4), (9,4)$.</li> <li>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 $(M,N)=(4,2)$.</li> <li>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. </li> </ol>