Gelfand-Ponomarev constructions for quadruples and sextuples, and Herrmann's endomorphisms

Rafael Stekolshchik

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Submission date: 28. Aug. 2006
Pages: 128
published in: Journal of pure and applied algebra, 211 (2007) 1, p. 95-202 
DOI number (of the published article): 10.1016/j.jpaa.2007.01.005
Bibtex
with the following different title: Gelfand-Ponomarev and Herrmann constructions for quadruples and sextuples
MSC-Numbers: 16G20, 06C05, 06B15
Keywords and phrases: Modular lattices, Perfect polynomials, Coxeter functor

Abstract:
The notions of a perfect element and an admissible element of

the free modular lattice generated by elements

are introduced by Gelfand and Ponomarev. We recall

that an element of a modular lattice L is said to be perfect,

if, for each finite dimension indecomposable K-linear representation

over any field K,

the image of a

is either zero, or , where is the

lattice of all vector K-subspaces of X.

Gelfand and Ponomarev gave a complete classification of such elements in the lattice , associated to the extended Dynkin diagram , and also in , where r > 4.

The main aim of this paper is to classify all the admissible

elements and all the perfect elements in the Dedekind lattice

generated by six elements that is associated to the extended Dynkin diagram . Gelfand and Ponomarev constructed

admissible elements of the lattice recurrently whereas we suggest a direct method for creating admissible elements. Using this method we also construct

admissible elements for and show that these elements coincide,

modulo linear equivalence, with admissible elements constructed

by Gelfand and Ponomarev. Admissible sequences and admissible elements

for (resp. ) form 14 classes (resp. 8 classes)

and possess a certain periodicity.

Our classification of perfect elements for is based on

the description of admissible elements. The constructed set

of perfect elements is the union of 64-element distributive

lattices , and is the distributive lattice itself.

The lattice of perfect elements obtained by Gelfand and

Ponomarev for can be imbedded into the lattice of perfect

elements , associated with .

Herrmann constructed perfect elements , ,

in by means of certain endomorphisms

and showed that these perfect elements coincide with

the Gelfand-Ponomarev perfect elements modulo linear equivalence.

We show that the admissible elements in are also obtained by

means of Herrmann's endomorphisms . Herrmann's

endomorphism and the elementary map of

Gelfand-Ponomarev act, in a sense, in opposite

directions, namely the endomorphism adds the index

to the beginning of the admissible sequence, and the elementary

map adds the index to the end of the admissible sequence.