Representation Theory of Combinatorial Categories.

[摘要] A representation V of a category D is a functor D --> Mod-R; the representations of D form an abelian category with natural transformations as morphisms. Say V is finitely generated if there exist finitely many vectors v_i in V d_i so that any strict subrepresentation of V misses some v_i. If every finitely generated representation satisfies both ACC and DCC on subrepresentations, we say D has dimension zero over R. The main theoretical result of this thesis is a practical recognition theorem for categories of dimension zero (Theorem 4.3.2). The main computational result is an algorithm for decomposing a finitely presented representation of a category of dimension zero into its multiset of irreducible composition factors (Theorem 4.3.5). Our main applications take D to be the category of finite sets; we explain how the general results of this thesis suggest specific experiments that lead to structure theory and practical algorithms in this case.

[发布机构] University of Michigan

[效力级别] Representation stability

[关键词] Representation theory of categories;Representation stability;Categories of dimension zero;Coherent functors;Mathematics;Science;Mathematics