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1. Unique Rectification in $d$-Complete Posets: Towards the $K$-Theory of Kac-Moody Flag Varieties

   https://doi.org/10.37236/7903  

   Ilango, Rahul ; Pechenik, Oliver ; Zlatin, Michael ( October 2018 , The Electronic Journal of Combinatorics)

   null (Ed.)

   The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of 'unique rectification targets' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $K$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's '$d$-complete posets' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of $d$-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain $K$-theoretic Schubert structure constants in the Kac-Moody setting. 

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2. Nonreciprocal directional dichroism of a chiral magnet in the visible range

   https://doi.org/10.1038/s41535-020-0224-6  

   Yokosuk, Michael O. ; Kim, Heung-Sik ; Hughey, Kendall D. ; Kim, Jaewook ; Stier, Andreas V. ; O’Neal, Kenneth R. ; Yang, Junjie ; Crooker, Scott A. ; Haule, Kristjan ; Cheong, Sang-Wook ; et al ( April 2020 , npj Quantum Materials)

   Nonreciprocal directional dichroism is an unusual light–matter interaction that gives rise to diode-like behavior in low-symmetry materials. The chiral varieties are particularly scarce due to the requirements for strong spin–orbit coupling, broken time-reversal symmetry, and a chiral axis. Here we bring together magneto-optical spectroscopy and first-principles calculations to reveal high-energy, broadband nonreciprocal directional dichroism in Ni₃TeO₆with special focus on behavior in the metamagnetic phase above 52 T. In addition to demonstrating this effect in the magnetochiral configuration, we explore the transverse magnetochiral orientation in which applied field and light propagation are orthogonal to the chiral axis and, by so doing, uncover an additional configuration with a unique nonreciprocal response in the visible part of the spectrum. In a significant conceptual advance, we use first-principles methods to analyze how the Ni²⁺d-to-don-site excitations develop magneto-electric character and present a microscopic model that unlocks the door to theory-driven discovery of chiral magnets with nonreciprocal properties.

    

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3. 2-Adic point counting on K3 surfaces

   https://doi.org/10.1007/s40993-022-00378-x  

   Elsenhans, Andreas-Stephan ; Jahnel, Jörg ( December 2022 , Research in Number Theory)

   Abstract This article reports on an approach to point counting on algebraic varieties over finite fields that is based on a detailed investigation of the 2-adic orthogonal group. Combining the new approach with a p -adic method, we count the number of points on some K 3 surfaces over the field $$\mathbb {F}_{\!p}$$ F p , for all primes $$p < 10^8$$ p < 10 8 . 

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4. Spectral Data for Spin Higgs Bundles

   https://doi.org/10.1093/imrn/rny296  

   Mukhopadhyay, Swarnava ; Wentworth, Richard ( January 2019 , International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and “twisted” spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program, which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spaces of projective orthogonal Higgs bundles (in the even case) and projective symplectic Higgs bundles (in the odd case). 

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5. Distribution of values of Gaussian hypergeometric functions

   https://doi.org/10.4310/PAMQ.2023.v19.n1.a14  

   Ono, Ken ; Saad, Hasan ; Saikia, Neelam ( January 2023 , Pure and Applied Mathematics Quarterly)

   In the 1980’s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Ap ́ery-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the 2F1 functions, the limiting distribution is semicircular (i.e. SU(2)), whereas the distribution for the 3F2 functions is the Batman distribution for the traces of the real orthogonal group O3. 

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