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1. A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

   https://doi.org/10.1007/s00029-024-00924-8  

   Lenart, Cristian; Naito, Satoshi; Sagaki, Daisuke (July 2024, Selecta Mathematica)

   NA (Ed.)

   We give a Chevalley formula for an arbitrary weight for the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory QK_{T}(G/B) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of QK_{T}(G/B). In type A_{n-1}, we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory QK(SL_{n}/B); we also obtain very explicit information about the coefficients in the respective Chevalley formula. 

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2. Fell’s absorption principle for semigroup operator algebras

   https://doi.org/10.4171/JNCG/566  

   Katsoulis, Elias G (April 2025, Journal of Noncommutative Geometry)

   Cuntz, J (Ed.)

   Fell’s absorption principle states that the left regular representation of a group absorbs any unitary representation of the group when tensored with it. In a weakened form, this result carries over to the left regular representation of a right LCM submonoid of a group and its Nica-covariant isometric representations but it fails if the semigroup does not satisfy independence. In this paper, we explain how to extend Fell’s absorption principle to an arbitrary submonoidPof a groupGby using an enhanced version of the left regular representation. Li’s semigroup\mathrm{C}^{*}-algebra\mathrm{C}^{*}_{s}(P)and its representations appear naturally in our context. Using the enhanced left regular representation, we not only provide a very concrete presentation for the reduced object for\mathrm{C}^{*}_{s}(P)but we also resolve open problems and obtain very transparent proofs of earlier results. In particular, we address the non-selfadjoint theory and we show that the non-selfadjoint object attached to the enhanced left regular representation coincides with that of the left regular representation. We obtain a non-selfadjoint version of Fell’s absorption principle involving the tensor algebra of a semigroup and we use it to improve recent results of Clouâtre and Dor-On on the residual finite dimensionality of certain\mathrm{C}^{*}-algebras associated with such tensor algebras. As another application, we give yet another proof for the existence of a\mathrm{C}^{*}-algebra which is co-universal for equivariant, Li-covariant representations of a submonoidPof a groupG. 

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3. Convergence of the environment seen from geodesics in exponential last-passage percolation

   https://doi.org/10.4171/JEMS/1594  

   Martin, James B; Sly, Allan; Zhang, Lingfu (March 2025, Journal of the European Mathematical Society)

   A well-known question in planar first-passage percolation concerns the convergence of the empirical distribution of weights as seen along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on\mathbb{Z}^{2}with i.i.d. exponential weights, and provide explicit formulae for the limiting distributions, which depend on the asymptotic direction. For example, for geodesics in the direction of the diagonal, the limiting weight distribution has density(1/4+x/2+x^{2}/8)e^{-x}, and so is a mixture of Gamma(1,1), Gamma(2,1), and Gamma(3,1) distributions with weights1/4,1/2, and1/4respectively. More generally, we study the local environment as seen from vertices along geodesics (including information about the shape of the path and about the weights on and off the path in a local neighborhood). We consider finite geodesics from(0,0)ton\boldsymbol{\rho}for some vector\boldsymbol{\rho}in the first quadrant, in the limit asn\to\infty, as well as semi-infinite geodesics in direction\boldsymbol{\rho}. We show almost sure convergence of the empirical distributions of the environments along these geodesics, as well as convergence of the distributions of the environment around a typical point in these geodesics, to the same limiting distribution, for which we give an explicit description.We make extensive use of a correspondence with TASEP as seen from an isolated second-class particle for which we prove new results concerning ergodicity and convergence to equilibrium. Our analysis relies on geometric arguments involving estimates for last-passage times, available from the integrable probability literature. 

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4. Mapping TASEP back in time

   https://doi.org/10.1007/s00440-021-01074-0  

   Petrov, Leonid; Saenz, Axel (July 2021, Probability Theory and Related Fields)

   Abstract We obtain a new relation between the distributions$$\upmu _t$$ ${\mu }_t$at different times$$t\ge 0$$ $t\ge 0$of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions$$\upmu _t$$ ${\mu }_t$backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving$$\upmu _t$$ ${\mu }_t$which in turn brings new identities for expectations with respect to$$\upmu _t$$ ${\mu }_t$. The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions. 

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5. Universal coarse geometry of spin systems

   https://doi.org/10.1007/s11005-025-01949-6  

   Elokl, Ali; Jones, Corey (June 2025, Letters in Mathematical Physics)

   Abstract The prospect of realizing highly entangled states on quantum processors with fundamentally different hardware geometries raises the question: to what extent does a state of a quantum spin system have an intrinsic geometry? In this paper, we propose that both states and dynamics of a spin system have a canonically associatedcoarse geometry, in the sense of Roe, on the set of sites in the thermodynamic limit. For a state$$\phi $$ $\varphi$on an (abstract) spin system with an infinite collection of sitesX, we define a universal coarse structure$$\mathcal {E}_{\phi }$$ $E_{\varphi }$on the setXwith the property that a state has decay of correlations with respect to a coarse structure$$\mathcal {E}$$ $E$onXif and only if$$\mathcal {E}_{\phi }\subseteq \mathcal {E}$$ $E_{\varphi }\subseteq E$. We show that under mild assumptions, the coarsely connected completion$$(\mathcal {E}_{\phi })_{con}$$ ${\left(E_{\varphi }\right)}_{\mathrm{con}}$is stable under quasi-local perturbations of the state$$\phi $$ $\varphi$. We also develop in parallel a dynamical coarse structure for arbitrary quantum channels, and prove a similar stability result. We show that several order parameters of a state only depend on the coarse structure of an underlying spatial metric, and we establish a basic compatibility between the dynamical coarse structure associated with a quantum circuit$$\alpha $$ $\alpha$and the coarse structure of the state$$\psi \circ \alpha $$ $\psi \circ \alpha$where$$\psi $$ $\psi$is any product state. 

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