Kleiner's theorem for unitary representations of posets

2011/03/05 by Yu. S. Samoĭlenko, Yurii Samoilenko, Samoilenko, Yurii +2
Mathematics · #16G20 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Primary 15A63 #Representation Theory (math.RT) #Secondary 15A21 #math.FA #math.RT #msc:15A21 #msc:15A63 #msc:16G20

paper · pdf · doi:10.48550/arxiv.1103.1085

12 pages, paper reorganized and rewritten. some statements were added

openalex publication_date 2011/03/05 · arxiv created 2012/02/18 · arxiv updated 2012/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A subspace representation of a poset \mathcal S=\s₁,...,s_t\ is given by a system (V;V₁,...,V_t) consisting of a vector space V and its subspaces V_i such that V_i⊆ V_j if s_i \prec s_j. For each real-valued vector χ=(χ₁,...,χ_t) with positive components, we define a unitary χ-representation of \mathcal S as a system (U;U₁,...,U_t) that consists of a unitary space U and its subspaces U_i such that U_i⊆ U_j if s_i\prec s_j and satisfies χ₁ P₁+...+χ_t P_t= \mathbb 1, in which P_i is the orthogonal projection onto U_i. We prove that \mathcal S has a finite number of unitarily nonequivalent indecomposable χ-representations for each weight χ if and only if \mathcal S has a finite number of nonequivalent indecomposable subspace representations; that is, if and only if \mathcal S contains any of Kleiner's critical posets.

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