Renormalization-Group Transformation in a $2n$-Component Fermionic Hierarchical Model

We study the $2N$-component fermionic model on a hierarchical lattice and give explicit formulas for the renormalization-group transformation in the space of coefficients that determine a Grassmann-valued density of the free measure. We evaluate the inverse renormalization-group transformation. The de.nition of the renormalization-group fixed points reduces to a solution of a system of algebraic equations. We investigate solutions of this system for $N=1,2,3$. For $\alpha=1$, we prove an analogue of the central limit theorem for fermionic $2N$-component fields. We discover an interesting relation between renormalization-group transformations in bosonic and fermionic hierarchical models and show that one of these transformations is obtained from the other by replacing $N$ with $-N$.