Gelfand–Ponomarev and Herrmann constructions for quadruples and sextuples

AbstractThe notions of a perfect element and an admissible element of the free modular lattice Dr 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 ρX:L→L(X) over any field K, the image ρX(a)⊆X of a is either zero, or ρX(a)=X, where L(X) is the lattice of all vector K-subspaces of X.A complete classification of such elements in the lattice D4, associated to the extended Dynkin diagram D˜4 (and also in Dr, 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 D2,2,2 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 Dr recurrently. We suggest a direct method for creating admissible elements. Using this method we also construct admissible elements for D4 and show that these elements coincide modulo linear equivalence with admissible elements constructed by Gelfand and Ponomarev. Admissible sequences and admissible elements for D2,2,2 (resp. D4) form 14 classes (resp. 8 classes) and possess some periodicity.Our classification of perfect elements for D2,2,2 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 D4 can be imbedded into the lattice of perfect elements H+, associated with D2,2,2.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 sn, tn, pi,n in D4 by means of some endomorphisms γij and showed that these perfect elements coincide with the Gelfand–Ponomarev perfect elements modulo linear equivalence. We show that the admissible elements in D4 are also obtained by means of Herrmann’s endomorphisms γij. Herrmann’s endomorphism γij and the elementary map of Gelfand–Ponomarev φi act, in a sense, in opposite directions, namely the endomorphism γij adds the index to the beginning of the admissible sequence, and the elementary map φi adds the index to the end of the admissible sequence