It is presented a complete categorical description of one of the main differentiation algorithms for representations of posets with involution — Differentiation II — constructed originally by the second author at the end of the 80's on the base of the matrix approach. A similar categorical description of some simpler additional differentiations is given as well.
The apparatus of differentiation DI-DV was introduced by A.G. Zavadskij to classify different kind of posets, in particular, Zavadskij et al. described categorical properties of algorithms of differentiation DI and DII as Gabriel did for the algorithm of differentiation with respect to a maximal point introduced by Nazarova and Roiter. In this paper, it is presented categorical properties of the algorithms of differentiation DIII, DIV and DV for posets with involution.
<abstract><p>Recently, Çanakçi and Schroll proved that associated with a string module $ M(w) $ there is an appropriated snake graph $ \mathscr{G} $. They established a bijection between the corresponding perfect matching lattice $ \mathscr{L}(\mathscr{G}) $ of $ \mathscr{G} $ and the canonical submodule lattice $ \mathscr{L}(M(w)) $ of $ M(w) $. We introduce Brauer configurations whose polygons are defined by snake graphs in line with these results. The developed techniques allow defining snake graphs, which after suitable procedures, build Kronecker modules. We compute the dimension of the Brauer configuration algebras and their centers arising from the different processes. As an application, we estimate the trace norm of the canonical non-regular Kronecker modules and some families of trees associated with some snake graphs classes.</p></abstract>
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