Separating functions, spectral theory of graphs, and locally scalar representations in Hilbert spaces

Abstract

We consider the connection of the separating functions ρ r with locally scalar representations of graphs and with spectral theory of graphs. Separating Functions ρ ρ ρ ρ rThe present paper is devoted to the investigation of the connection of the separating functions ρ r introduced in [1] with locally scalar representations of graphs, on the one hand, and with spectral theory of graphs, on the other.A function P ( S ) that associates a partially ordered set S with a positive rational number was introduced in [2]. In terms of the function P, a criterion for the finite representability and tameness of an arbitrary partially ordered set was formulated in [2]: a partially ordered set S is representable (tame) if and only if P ( S ) < 4 ( P ( S ) = 4). If a partially ordered set S is the union of s disjoint chains (i.e., chains such that any two elements belonging to different chains are incomparable), then P ( S ) = ρ ( n 1 , n 2 , … , n s ) = i s i n = ∑ 1 ρ( ) = i s = ∑ 1

On functions on graphs and representations of a certain class of ∗-algebras

“…Formulas for such an evolution for special classes of graphs were obtained in [12]. We essentially follow [18] (a different approach was independently used in [13] to obtain similar results).…”

“…Following [12] we decompose any p-invariant weight into odd and even parts and study their evolution under the Coxeter functors (Section 3.3); similar formulas have been independently obtained in [13] in different terms. The obtained formulas are used in Section 3.5 to prove that for any graph containing an extended Dynkin graph as a proper subgraph there exists a weight such that the corresponding algebra has infinite-dimensional irreducible * -representations (Theorem 20).…”