Gelfand–Ponomarev and Herrmann constructions for quadruples and sextuples

Abstract

The notions of a perfect element and an admissible element of the free modular lattice D r generated by r ≥ 1 elements are introduced by Gelfand and Ponomarev in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii (1901–1973), IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3–58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1–56]. We recall that an element a ∈ D of a modular lattice L is perfect, if for each finite-dimension indecomposable K -linear representation ρ X : L → L ( X ) over any field K , the image ρ X ( a ) ⊆ X of a is either zero, or ρ X ( a ) = X , where L ( X ) is the lattice of all vector K -subspaces of X . A complete classification of such elements in the lattice D 4 , associated to the extended Dynkin diagram D ˜ 4 (and also in D r , where r > 4 ) is given in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii (1901–1973), IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3–58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1–56; I.M. Gelfand, V.A. Ponomarev, Lattices, representations, and their related algebras, I, Uspehi Mat. Nauk 31 (5(191)) (1976) 71–88 (Russian); English translation: Russian Math. Surv. 31 (5) (1976) 67–85; I.M. Gelfand, V.A. Ponomarev, Lattices, representations, and their related algebras, II. Uspehi Mat. Nauk 32 (1(193)) (1977) 85–106 (Russian); English translation: Russian Math. Surv. 32 (1) (1977) 91–114]. The main aim of the present paper is to classify all the admissible elements and all the perfect elements in the Dedekind lattice D 2 , 2 , 2 generated by six elements that are associated to the extended Dynkin diagram E ˜ 6 . We recall that in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii (1901–1973), IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3–58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1–56], Gelfand and Ponomarev construct admissible elements of the lattice D r recurrently. We suggest a direct method for creating admissible elements. Using this method we also construct admissible elements for D 4 and show that these elements coincide modulo linear equivalence with admissible elements constructed by Gelfand and Ponomarev. Admissible sequences and admissible elements for D 2 , 2 , 2 (resp. D 4 ) form 14 classes (resp. 8 classes) and possess some periodicity. 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 . Herrmann in [C. Herrmann, Rahmen und erzeugende Quadrupel in modularen Verbänden. (German) [Frames and generating quadruples in modular lattices], Algebra Universalis 14 (3) (1982) 357–387] constructed perfect elements s n , t n , p i , n in D 4 by means of some endomorphisms γ i j 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 i j . Herrmann’s endomorphism γ i j and the elementary map of Gelfand–Ponomarev φ i act, in a sense, in opposite directions, namely the endomorphism γ i j adds the index to the beginning of the admissible sequence, and the elementary map φ i adds the index to the end of the admissible sequence.