Singular locally scalar representations of quivers in Hilbert spaces and separating functions

Redchuk, I. K.; Roiter, A. V.

doi:10.1007/pl00022183

Cited by 4 publications

(7 citation statements)

References 6 publications

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“…The present paper is dedicated to studying the connections of separating functions ρ r , introduced in [1], with locally scalar representations of graphs on the one hand, and with spectral graph theory on the other hand.…”

Section: Separating Functions ρ Rmentioning

confidence: 99%

“…In [1] the natural generalization of the function ρ was suggested. Given m ∈ N, denote V m the set of unordered suits of m nonnegative integers; then denote V = m∈N V m .…”

Section: Separating Functions ρ Rmentioning

confidence: 99%

“…The functions ρ r was exploited for describing the standard characters of locally scalar representations of certain types of graphs (see [1], [5]).…”

Section: Separating Functions ρ Rmentioning

confidence: 99%

“…In [1] the explicit formulas in terms of the functions ρ r , indicating how odd and even standard vectors transform under the action of c t , was obtained for certain graphs of the special type. Now we will prove an analogous statement for an arbitrary bipartite graph G. Proposition 3.1.…”

Section: Standard Characters Of Star-shaped Graphsmentioning

confidence: 99%

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

Redchuk

2006

Ukr Math J

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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

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Section: Separating Functions ρ Rmentioning

confidence: 99%

“…In [1] the natural generalization of the function ρ was suggested. Given m ∈ N, denote V m the set of unordered suits of m nonnegative integers; then denote V = m∈N V m .…”

Section: Separating Functions ρ Rmentioning

confidence: 99%

“…The functions ρ r was exploited for describing the standard characters of locally scalar representations of certain types of graphs (see [1], [5]).…”

Section: Separating Functions ρ Rmentioning

confidence: 99%

Section: Standard Characters Of Star-shaped Graphsmentioning

confidence: 99%

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

Redchuk

2006

Ukr Math J

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“…Let us recall a few facts about root systems associated with extended Dynkin graphs (see [1], [15]). Let G be a simple connected graph.…”

Section: Introductionmentioning

confidence: 99%

The Spectral Problem and Algebras Associated with Extended Dynkin Graphs

Popovych

2009

Preprint

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The Spectral Problem is to describe possible spectra σ(A j ) for an irreducible n-tuple of Hermitian operators s.t. A 1 + . . . + A n is a scalar operator. In case when m j = |σ(A j )| are finite and a rooted tree T m 1 ,...,mn with n branches of lengths m 1 , . . . , m n is a Dynkin graph the explicit answer to the Spectral Problem was given recently by S. A. Kruglyak, S. V. Popovych, and Yu. S. Samoǐlenko in [6]. In present work the solution of the Spectral Problem for all star-shaped simply laced extended Dynkin graphs, i.e. when (m 1 , . . . , m n ) ∈ {(2, 2, 2, 2), (3, 3, 3), (4, 4, 2), (6, 3, 2)} is presented.

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The Smith normal form of the walk matrix of the extended Dynkin graph D˜n

Moon,

Park

2023

Linear Algebra and its Applications

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Singular locally scalar representations of quivers in Hilbert spaces and separating functions

Cited by 4 publications

References 6 publications

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

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

The Spectral Problem and Algebras Associated with Extended Dynkin Graphs

The Smith normal form of the walk matrix of the extended Dynkin graph D˜n

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