## Algebraic analysis of the Rarita--Schwinger system in real dimension three

*Alberto Damiano*

**Address.**

Mathematics Institute, Charles University, Sokolovska 83, Prague, Czech Republic

**E-mail. **

damiano@karlin.mff.cuni.cz

**Abstract.**

In this paper we use the explicit description of the
Spin--$\frac{3}{2}$ Dirac operator in real dimension $3$ appeared in \cite{homma} to perform the algebraic analysis of the space of nullsolution of the system of equations given by several Rarita--Schwinger operators. We make use of the general theory provided by \cite{csss1} and some standard Gr\"obner Bases techniques. Our aim is to show
that such operator shares many of the algebraic properties of the Dirac operator in real dimension four. In particular, we prove the exactness of the associated algebraic complex, a duality result and we explicitly describe the space of polynomial solutions.