Characterizing partition functions of the vertex model

Draisma, Jan; Gijswijt, Dion; Lovász, László; Regts, Guus; Schrijver, Alexander

doi:10.1016/j.jalgebra.2011.10.030

Cited by 22 publications

(34 citation statements)

References 10 publications

(20 reference statements)

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“…Section 4 deals with some framework, which we need to prove Theorem 1. This framework is similar to the framework developed in [5]. In particular, the connection with the invariant theory of the symplectic group will become clear there.…”

Section: Introductionmentioning

confidence: 68%

“…Theorem 1.) We note that no matter what value we choose for , this f is never the (ordinary) partition function of an edge-coloring model, as follows from the results in [5], see also [17,Proposition 5.6]. It turns out that skew-partition functions are graph parameters satisfying the modified conditions in (2), and moreover, these are the only graph parameters.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Graph parameters from symplectic group invariants

Regts

Sevenster²

2017

Journal of Combinatorial Theory, Series B

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Section: Introductionmentioning

confidence: 68%

Section: Introductionmentioning

confidence: 98%

Graph parameters from symplectic group invariants

Regts

Sevenster²

2017

Journal of Combinatorial Theory, Series B

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“…Then the first inequality in (8) follows from the fact that π S,U (C S ) is a GL(V )-orbit in F U , and that C U has minimal dimension among all GL(V )-orbits in F U . By the maximality of dim(C U ), we have equality throughout (8). As π S,U (C S ) is a GL(V )-orbit and as C U is the unique orbit in F U of minimal dimension, (7) follows.…”

Section: Proof Of Theorems 1 Andmentioning

confidence: 97%

“…Thus we deal with invariants of ordinary directed graphs, with no ordering of edges. This case was considered in [8], and Theorem 1 forms a generalization of its result.…”

Section: Examples (Continued)mentioning

confidence: 99%

On traces of tensor representations of diagrams

Schrijver

2015

Linear Algebra and its Applications

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Let T be an (abstract) set of types, and let ι, o : T → Z + .A T -diagram is a locally ordered directed graph G equipped with a function τ : V (G) → T such that each vertex v of G has indegree ι(τ (v)) and outdegree o(τ (v)). (A directed graph is locally ordered if at each vertex v, linear orders of the edges entering v and of the edges leaving v are specified.) Let V be a finite-dimensional F-linear space, where F is an algebraically closed field of characteristic 0. A function R on T assigning to each t ∈ T a tensor R(t) ∈ V * ⊗ι(t) ⊗ V ⊗o(t) is called a tensor representation of T . The trace (or partition function) of R is the F-valued function p R on the collection of T -diagrams obtained by 'decorating' each vertex v of a T -diagram G with the tensor R(τ (v)), and contracting tensors along each edge of G, while respecting the order of the edges entering v and leaving v. In this way we obtain a tensor network. We characterize which functions on T -diagrams are traces, and show that each trace comes from a unique 'strongly nondegenerate' tensor representation. The theorem applies to virtual knot diagrams, chord diagrams, and group representations.

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“…Furthermore, a 'k-join lemma' is given below that simplifies the proof. The complex case, as studied in [4], [21], demands different conditions and machinery, and requires (so far) the dimension of the vertex model to be specified in the theorem.…”

Section: Corollarymentioning

confidence: 99%

On partition functions for 3-graphs

Regts

Schrijver

Sevenster

2016

Journal of Combinatorial Theory, Series B

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Abstract.A cyclic graph is a graph with at each vertex a cyclic order of the edges incident with it specified. We characterize which real-valued functions on the collection of cubic cyclic graphs are partition functions of a real vertex model (P. de la Harpe, V.F.R. Jones, Graph invariants related to statistical mechanical models: examples and problems, Journal of Combinatorial Theory, Series B 57 (1993) 207-227). They are characterized by 'weak reflection positivity', which amounts to the positive semidefiniteness of matrices based on the 'k-join' of cubic cyclic graphs (for all k ∈ Z + ).Basic tools are the representation theory of the symmetric group and geometric invariant theory, in particular the Hanlon-Wales theorem on the decomposition of Brauer algebras and the ProcesiSchwarz theorem on inequalities defining orbit spaces.

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Characterizing partition functions of the vertex model

Cited by 22 publications

References 10 publications

Graph parameters from symplectic group invariants

Graph parameters from symplectic group invariants

On traces of tensor representations of diagrams

On partition functions for 3-graphs

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