On Admissible and Perfect Elements in the Modular Lattice

Abstract

For the modular lattice D^4 = {1+1+1+1} associated with the extended Dynkin diagram \tilde{D}_4 (and also for D^r, where r > 4), Gelfand and Ponomarev introduced the notion of admissible and perfect lattice elements and classified them. In this work, we classify the admissible and perfect elements in the modular lattice D^{2,2,2} = {2+2+2} associated with the extended Dynkin diagram \tilde{E}_6. Gelfand and Ponomarev constructed admissible elements for D^r recurrently in the length of multi-indices, which they called admissible sequences. Here we suggest a direct method for creating admissible elements. Admissible sequences and admissible elements for D^{2,2,2} (resp. D^4) form 14 classes (resp. 11 classes) and possess some periodicity. If under all indecomposable representations of a modular lattice the image of an element is either zero or the whole representation space, the element is said to be perfect. Our classification of perfect elements for D^{2,2,2} is based on the description of admissible elements. The constructed set H^+ of perfect elements is the union of 64-element distributive lattices H^+(n), and H^+ is the distributive lattice itself. The lattice of perfect elements B^+ obtained by Gelfand and Ponomarev for D^4 can be imbedded into the lattice of perfect elements H^+, associated with D^{2,2,2}.