Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Internal edge vectors on plabic networks in the disk and a
  generalization of Talaska formula   Following [42], positroid cells ${\mathcal S}_{\mathcal M}^{\mbox{TNN}}$ in
totally non-negative Grassmannians ${Gr^{\mbox{TNN}} (k,n)}$ admit
parametrizations by positive weights on planar bicolored directed perfect
networks in the disk. An explicit formula for elements of matrices representing
the points in ${\mathcal S}_{\mathcal M}^{\mbox{TNN}}$ was obtained in [49] in
terms of flows on such networks. The formulas from [42,49] are defined on the
boundary edge vectors. In this paper we propose an extension of these formulas
for vectors on internal edges defined as summations over paths on the given
directed network gauged by the choice of a ray direction. This gauge choice
does not affect the boundary edge vectors, which generate the Postnikov
boundary measurement map. The systems of internal edge vectors corresponding to
different choices of gauge ray directions coincide up to sign, the sign rule
admits a simple explicit description. We prove that the components of these
edge vectors are rational in the weights with subtraction--free denominators.
Moreover, these components are expressed in terms of internal edge flows; these
formulas extend the original Talaska ones to the internal edges. These vectors
also solve the system of geometric relations associated to the corresponding
network. These relations are full rank and respect the total non--negativity
property on the full positroid cell. We also provide explicit formulas both for
the transformation rules of the edge vectors with respect to the orientation,
and for their transformations due to moves and reductions of networks.

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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: patt-sol