Preprint 92/2006

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
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

The notions of a perfect element and an admissible element of

the free modular lattice formula26 generated by formula28 elements

are introduced by Gelfand and Ponomarev. We recall

that an element formula30 of a modular lattice L is said to be perfect,

if, for each finite dimension indecomposable K-linear representation

formula36 over any field K,

the image formula40 of a

is either zero, or formula44, where formula46 is the

lattice of all vector K-subspaces of X.

Gelfand and Ponomarev gave a complete classification of such elements in the lattice formula52, associated to the extended Dynkin diagram formula54 , and also in formula26, 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 formula60

generated by six elements that is associated to the extended Dynkin diagram formula62. Gelfand and Ponomarev constructed

admissible elements of the lattice formula26 recurrently whereas we suggest a direct method for creating admissible elements. Using this method we also construct

admissible elements for formula52 and show that these elements coincide,

modulo linear equivalence, with admissible elements constructed

by Gelfand and Ponomarev. Admissible sequences and admissible elements

for formula60 (resp. formula52) form 14 classes (resp. 8 classes)

and possess a certain periodicity.

Our classification of perfect elements for formula60 is based on

the description of admissible elements. The constructed set formula78

of perfect elements is the union of 64-element distributive

lattices formula82, and formula78 is the distributive lattice itself.

The lattice of perfect elements formula86 obtained by Gelfand and

Ponomarev for formula52 can be imbedded into the lattice of perfect

elements formula78, associated with formula60.

Herrmann constructed perfect elements formula94, formula96,

formula98 in formula52 by means of certain endomorphisms

formula102 and showed that these perfect elements coincide with

the Gelfand-Ponomarev perfect elements modulo linear equivalence.

We show that the admissible elements in formula52 are also obtained by

means of Herrmann's endomorphisms formula102. Herrmann's

endomorphism formula102 and the elementary map of

Gelfand-Ponomarev formula110 act, in a sense, in opposite

directions, namely the endomorphism formula102 adds the index

to the beginning of the admissible sequence, and the elementary

map formula110 adds the index to the end of the admissible sequence.

23.06.2018, 02:11