Let $ \left( X^3,g \right)$ be a closed, oriented Riemannian $3$-manifold. Choose a spin structure, and denote by $\mathbb S $ the spinor bundle and $\mathbf c $ the Clifford multiplication.
The Dirac operator is obtained as follows. Take the Levi-Civita connection on $\mathbb S$ . That's a first order differential operator from sections of $\mathbb S$ to sections of $T^* X \otimes \mathbb S $. Then we compose by Clifford multiplication seen as a bundle map $T^* X \otimes \mathbb S \to \mathbb S$.
I have come across a similar constructions for symmetric spinors. Namely, denote by $\mathbb S^k $ the $k$th symmetric power of $\mathbb S $. This is the bundle associated to the rank $k+1$ irreducible representation of $\text{SU} (2)$. As such, it inherits a spin connection. Moreover, Clifford multiplication induces a bundle map $T^*X \otimes \mathbb S^k \to \mathbb S^k $ obtained by the restriction of $\mathbf c \otimes 1 \otimes \cdots \otimes 1 + \cdots 1 \otimes \cdots \otimes 1 \otimes \mathbf c $. This defines a first order operator $D$ on sections of $\mathbb S ^k$.
I'd like to know as much as possible about these operators. They are so natural that I am sure they have been studied extensively, I just don't know their name. I am particularly interested in the case $k$ even (and in particular $k=4$) because in that case the operator we obtain does not depend on the spin structure.
Could you please give me a reference where these kind of operators are studied? Thank you very much.