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Critical Varieties in the Grassmannian

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Abstract

We introduce a family of spaces called critical varieties. Each critical variety is a subset of one of the positroid varieties in the Grassmannian. The combinatorics of positroid varieties is captured by the dimer model on a planar bipartite graph G, and the critical variety is obtained by restricting to Kenyon’s critical dimer model associated to a family of isoradial embeddings of G. This model is invariant under square/spider moves on G, and we give an explicit boundary measurement formula for critical varieties which does not depend on the choice of G. This extends our recent results for the critical Ising model, and simultaneously also includes the case of critical electrical networks. We systematically develop the basic properties of critical varieties. In particular, we study their real and totally positive parts, the combinatorics of the associated strand diagrams, and introduce a shift map motivated by the connection to zonotopal tilings and scattering amplitudes.

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Notes

  1. Strictly speaking, we have \({\text {wt}}_{\varvec{\theta }}(e)=\frac{1}{2i}{\text {wt}}_{\textbf{t}}(e)\) for all non-boundary \(e\in E(G)\), but this rescaling does not affect the boundary measurements since \({\text {wt}}_{\textbf{t}}\) and \({\text {wt}}_{\varvec{\theta }}\) are gauge-equivalent; see the proof of Lemma 5.9.

  2. By linearity of Ohm’s and Kirchhoff’s laws, knowing \(\Lambda ^{{\mathbb {T}}}_{p,q}\) for all \(1\leqslant p,q\leqslant N\) allows one to solve the more general problem: for any known voltages that are applied to the boundary vertices, one finds the resulting currents flowing through each boundary vertex.

  3. More precisely, Lam’s embedding lands in \({\text {Gr}}_{\geqslant 0}(N-1,2N)\). To get an element of \({\text {Gr}}_{\geqslant 0}(N+1,2N)\), one needs to apply the duality discussed in Sect. 3.2.

  4. The point \(X_0^{({k,n})}\) is the unique element of \({\text {Gr}}_{\geqslant 0}(k,n)\) satisfying \(S(X_0^{({k,n})})=X_0^{({k,n})}\), where \(S:{\text {Gr}}_{\geqslant 0}(k,n)\rightarrow {\text {Gr}}_{\geqslant 0}(k,n)\) is the cyclic shift automorphism discussed in Sect. 3.1.

  5. When (un)contracting a degree 2 interior vertex, we always assume that both edges incident to it have weight 1; this is always achievable by applying gauge transformations.

  6. We thank Lauren Williams for comments motivating the below results.

  7. The name is explained by the fact that if \({\varvec{\theta }}\) and \({\textbf{t}}\) are related by (5.1) then up to a simple transformation, the rows of \(F_{f,{\textbf{t}}}\) yield the Fourier coefficients of \(\varvec{\gamma }_{f,{\varvec{\theta }}}(t)\), viewed as a \(2\pi \)-periodic function of t.

  8. In order to invoke [OPS15, Proposition 9.8] one needs to assume that the Grassmann necklace of f is connected. However, even when it is not connected, the Grassmann necklace curve may be deformed slightly into a simple closed curve surrounding \(\Sigma _k(G)\); see the discussion around [BW20, Definition 4.4].

  9. The names (M1), (M2), (M3) for the moves in Fig. 23 are taken from [Gal18]: these moves are obtained from a single transformation of 3-dimensional zonotopal tilings by taking horizontal sections by planes \(z=k\) for \(k=1,2,3\), respectively; see [Gal18, Figure 8].

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Acknowledgements

I am indebted to Pasha Pylyavskyy for his numerous contributions at various stages of the development of [Gal22], where the boundary measurement formula was first discovered in the context of the Ising model. The generalization to the Grassmannian level was inspired by the results of [CLR20, KLRR18], presented by Marianna Russkikh at the “Dimers in Combinatorics and Cluster Algebras” conference at the University of Michigan. I thank Marianna for bringing these results to my attention, and also thank the organizers of the conference (Sebastian Franco, Gregg Musiker, Richard Kenyon, David Speyer, and Lauren Williams) for making such an interaction possible. Finally, I am grateful to Lauren Williams and to the anonymous referee for their valuable comments on the first version of the text.

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  1. Department of Mathematics, University of California, Los Angeles, CA, 90095, USA

    Pavel Galashin

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P. G. was supported by an Alfred P. Sloan Research Fellowship and by the National Science Foundation under Grants No. DMS-1954121 and No. DMS-2046915.

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Galashin, P. Critical Varieties in the Grassmannian. Commun. Math. Phys. 401, 3277–3333 (2023). https://doi.org/10.1007/s00220-023-04718-5

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