Gelfand-Ponomarev constructions for quadruples and sextuples, and Herrmann's endomorphisms
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Submission date: 28. Aug. 2006
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 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.