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  1. ; ; ; (, Proceedings of the American Mathematical Society)
    In this paper we define an interesting family of perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two with trivial multiplication on the $$ \operatorname {Tor}$$ algebra. This family is likely to play a key role in classifying perfect ideals with five generators of type two. 
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  2. ; (, Daehan suhaghoe nonmunjib)
    In this paper we give some branching rules for the fundamental representations of Kac--Moody Lie algebras associated to $$T$$-shaped graphs. These formulas are useful to describe generators of the generic rings for free resolutions of length three described in \cite{JWm18}. We also make some conjectures about the generic rings. 
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  3. ; (, Transactions of the American Mathematical Society)
    We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type $$ \mathbb{A}$$, we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type $$ \mathbb{A}$$ quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation. 
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