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

Rafael Stekolshchik


The notions of a perfect element and an admissible element of the free modular lattice $D^r$ generated by $r \geq 1$ elements are introduced by Gelfand and Ponomarev. We recall that an element $a \in D$ of a modular lattice $L$ is said to be {\it perfect}, if, for each finite dimension indecomposable $K$-linear representation $\rho_X : L \rightarrow \mathcal{L}(X)$ over any field $K$, the image $\rho_X(a) \subseteq X$ of $a$ is either zero, or $\rho_X(a) = X$, where $\mathcal{L}(X)$ is the lattice of all vector $K$-subspaces of $X$.

Gelfand and Ponomarev gave a complete classification of such elements in the lattice $D^4$, associated to the extended Dynkin diagram $\widetilde{\mathbb{D}}_4$, and also in $D^r$, where $r > 4$.

The main aim of this paper is to classify all the {\it admissible elements} and all the perfect elements in the Dedekind lattice $D^{2,2,2}$ generated by six elements that is associate to the extended Dynkin diagram $\widetilde{\mathbb{E}}_6$. Gelfand and Ponomarev constructed admissible elements of the lattice $D^r$ recurrently whereas we suggest a direct method for creatin 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 a certain 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 constructed perfect elements $s_n$, $t_n$, $p_{i,n}$ in $D^4$ by means of certain endomorphisms $\gamma_{ij}$ 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 $\gamma_{ij}$. Herrmann's endomorphism $\gamma_{ij}$ and the {\it elementary map} of Gelfand-Ponomarev $\phi_i$ act, in a sense, in opposite directions, namely the endomorphism $\gamma_{ij}$ adds the index to the beginning of the admissible sequence, and the elementary map $\phi_i$ adds the index to the end of the admissible sequence.

Aug 28, 2006
Aug 28, 2006
MSC Codes:
16G20, 06C05, 06B15
Modular lattices, Perfect polynomials, Coxeter functor

Related publications

2007 Repository Open Access
Rafael Stekolshchik

Gelfand-Ponomarev and Herrmann constructions for quadruples and sextuples

In: Journal of pure and applied algebra, 211 (2007) 1, pp. 95-202