Quadruples, admissible elements and Herrmann's endomorphisms

Abstract

We obtain a connection between admissible elements for quadruples and Herrmann's endomorphisms. Herrmann constructed perfect elements $s_n$, $t_n$, $p_{i,n}$ in $D^4$ by means of some endomorphisms and showed that these perfect elements coincide with the Gelfand-Ponomarev perfect elements modulo linear equivalence. We show that the admissible elements in $D^4$ are also obtained by means of Herrmann's endomorphisms $γ_{ij}$. Endomorphism $γ_{ij}$ and the elementary map of Gelfand-Ponomarev $ϕ_i$ act, in a sense, in opposite directions, namely the endomorphism $γ_{ij}$ adds the index to the start of the admissible sequence, and the elementary map $ϕ_i$ adds the index to the end of the admissible sequence.