# Article

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Summary:
In this paper we use the explicit description of the Spin–$\frac{3}{2}$ Dirac operator in real dimension $3$ appeared in (Homma, Y., The Higher Spin Dirac Operators on $3$–Dimensional Manifolds. Tokyo J. Math. 24 (2001), no. 2, 579–596.) 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 (Colombo, F., Sabadini, I., Sommen, F., Struppa, D. C., Analysis of Dirac systems and computational algebra, Progress in Mathematical Physics, Vol. 39, Birkhäuser, Boston, 2004.) and some standard Gröbner 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.
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