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1. The Positive Tropical Grassmannian, the Hypersimplex, and the m = 2 Amplituhedron

   https://doi.org/10.1093/imrn/rnad010  

   Łukowski, Tomasz; Parisi, Matteo; Williams, Lauren K (March 2023, International Mathematics Research Notices)

   Abstract The positive Grassmannian $$Gr^{\geq 0}_{k,n}$$ is a cell complex consisting of all points in the real Grassmannian whose Plücker coordinates are non-negative. In this paper we consider the image of the positive Grassmannian and its positroid cells under two different maps: the moment map$$\mu $$ onto the hypersimplex [ 31] and the amplituhedron map$$\tilde{Z}$$ onto the amplituhedron [ 6]. For either map, we define a positroid dissection to be a collection of images of positroid cells that are disjoint and cover a dense subset of the image. Positroid dissections of the hypersimplex are of interest because they include many matroid subdivisions; meanwhile, positroid dissections of the amplituhedron can be used to calculate the amplituhedron’s ‘volume’, which in turn computes scattering amplitudes in $$\mathcal{N}=4$$ super Yang-Mills. We define a map we call T-duality from cells of $$Gr^{\geq 0}_{k+1,n}$$ to cells of $$Gr^{\geq 0}_{k,n}$$ and conjecture that it induces a bijection from positroid dissections of the hypersimplex $$\Delta _{k+1,n}$$ to positroid dissections of the amplituhedron $$\mathcal{A}_{n,k,2}$$; we prove this conjecture for the (infinite) class of BCFW dissections. We note that T-duality is particularly striking because the hypersimplex is an $(n-1)$-dimensional polytope while the amplituhedron $$\mathcal{A}_{n,k,2}$$ is a $2k$-dimensional non-polytopal subset of the Grassmannian $$Gr_{k,k+2}$$. Moreover, we prove that the positive tropical Grassmannian is the secondary fan for the regular positroid subdivisions of the hypersimplex, and prove that a matroid polytope is a positroid polytope if and only if all 2D faces are positroid polytopes. Finally, toward the goal of generalizing T-duality for higher $$m$$, we define the momentum amplituhedron for any even $$m$$. 

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2. A Pattern Avoidance Characterization for Smoothness of Positroid Varieties

   Billey, Sara C.; Weaver, Jordan E. (January 2022, Séminaire lotharingien de combinatoire)

   Positroids are certain representable matroids originally studied by Post- nikov in connection with the totally nonnegative Grassmannian and now used widely in algebraic combinatorics. The positroids give rise to determinantal equations defin- ing positroid varieties as subvarieties of the Grassmannian variety. Rietsch, Knutson– Lam–Speyer and Pawlowski studied geometric and cohomological properties of these varieties. In this paper, we continue the study of the geometric properties of positroid varieties by establishing several equivalent conditions characterizing smooth positroid varieties using a variation of pattern avoidance defined on decorated permutations, which are in bijection with positroids. Furthermore, we give a combinatorial method for determining the dimension of the tangent space of a positroid variety at key points using an induced subgraph of the Johnson graph. We also give a Bruhat interval char- acterization of positroids. 

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3. Feigin–Odesskii brackets associated with Kodaira cycles and positroid varieties

   https://doi.org/10.1112/plms.70054  

   Hua, Zheng; Polishchuk, Alexander (May 2025, Proceedings of the London Mathematical Society)

   Abstract We establish a link between open positroid varieties in the Grassmannians and certain moduli spaces of complexes of vector bundles over Kodaira cycle , using the shifted Poisson structure on the latter moduli spaces and relating them to the standard Poisson structure on . This link allows us to solve a classification problem for extensions of vector bundles over . Based on this solution we further classify the symplectic leaves of all positroid varieties in with respect to the standard Poisson structure. Moreover, we get an explicit description of the moduli stack of symplectic leaves of with the standard Poisson structure as an open substack of the stack of vector bundles on . 

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4. Positroid Catalan numbers

   https://doi.org/10.1090/cams/33  

   Galashin, Pavel; Lam, Thomas (April 2024, Communications of the American Mathematical Society)

   Given a (bounded affine) permutation $f$, we study thepositroid Catalan number $C_f$defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class ofrepetition-free permutationsand show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated $q,t$-polynomials coincide with thegeneralized $q,t$-Catalan numbersthat recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov–Rozansky homology of Coxeter links. 

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5. Regularity theorem for totally nonnegative flag varieties

   https://doi.org/10.1090/jams/983  

   Galashin, Pavel; Karp, Steven; Lam, Thomas (April 2022, Journal of the American Mathematical Society)

   We show that the totally nonnegative part of a partial flag variety G / P G/P (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov. 

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