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The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart– Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang–Baxter moves, which biject the objects of the models associated to two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang–Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula for the equivariant K-group of semi-infinite flag manifolds. The generalized quantum Yang–Baxter moves give rise to a “sijection” (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula. [ABSTRACT FROM AUTHOR]

Let p be a prime, k be a p-closed field of characteristic different from p, and 1 → T → G → F → 1 be an exact sequence of algebraic groups over k, where T is a torus and F is a finite p-group. In this paper, we study the essential dimension ed(G; p) of G at p. R. Lötscher, M. MacDonald, A. Meyer, and the first author showed that min dim(V) - dim(G) 6 ed(G; p) 6 min dim(W) - dim(G); where V and W range over the p-faithful and p-generically free k-representations of G, respectively. In the special case where G = F, one recovers the formula for ed(F; p) proved earlier by N. Karpenko and A. Merkurjev. In the case where F = T, one recovers the formula for ed(T; p) proved earlier by R. Lötscher et al. In both of these cases, the upper and lower bounds on ed(G; p) given above coincide. In general, there is a gap between them. Lötscher et al. conjectured that the upper bound is, in fact, sharp; that is, ed(G; p) = min dim(W) - dim(G), where W ranges over the p-generically free representations. We prove this conjecture in the case where F is diagonalizable. [ABSTRACT FROM AUTHOR]

In this paper, we give a counterexample to Zariski's cancellation problem for principal ${\mathbb{G}}_{a}$ G a -bundles over certain open subvarieties of non- $\mathbb{A}^{1}$ A 1 -uniruled affine varieties. [ABSTRACT FROM AUTHOR]

Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category \mathcal {O} of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our representation-theoretical construction is based on the study of the action of Cartan-Drinfeld subalgebras. We prove the algebraic stable maps are invertible and depend rationally on the spectral parameter. As an application, we obtain new R-matrices in the category \mathcal {O} and we establish that a large family of simple modules, including the prefundamental representations associated to Q-operators, generically commute as representations of the Cartan-Drinfeld subalgebra. We also establish categorified QQ^*-systems in terms of the R-matrices we construct. [ABSTRACT FROM AUTHOR]

It is essential to guarantee product quality by detecting defects in industrial printed labels. Defect detection based on reference comparison is a common method to achieve this task. However, this method yields poor performance under large deformations of printed labels, due to the lack of accurate registration ability between testing images and reference images. Therefore, it is an urgent task to achieve accurate image registration for printed label defect detection. In this paper, a coarse-to-fine end-to-end registration network, named APPR-Net, is proposed to deal with the large deformation. First, we adopt a strategy of image patch splitting and stitching to improve the scalability of image resolution. Second, we design a four-stream affine transformation module followed by a multi-scale pyramid registration network, where a coarse registration is obtained by the former module and then gradually refined by the later network in a coarse-to-fine manner. Third, we introduce a distortion loss function to constrain the text distortion of the warped image after image registration. Finally, to simulate printed labels with defects and large deformation, we build a synthetic database based on real-world industrial printed labels for performance comparison. The results demonstrate that the proposed APPR-Net significantly outperforms other compared methods. • Propose a coarse-to-fine registration network to deal with large deformation. • Take advantages of image patch splitting and stitching strategy. • Design a four-stream affine transformation module followed by a pyramid network. • Introduce a distortion loss function to constrain text distortions. • Build a synthetic database based on real-world printed labels. [ABSTRACT FROM AUTHOR]

Recently, Blasiak–Morse–Seelinger introduced symmetric func- tions called Katalan functions, and proved that the K-theoretic k-Schur functions due to Lam–Schilling–Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called closed k-Schur Katalan functions is identified with the Schubert structure sheaves in the K-homology of the affine Grassmannian. Our main result is a proof of this conjecture. We also study a K-theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a nongeometric manner, based on the unipotent solution of the relativistic Toda lattice of Ruijsenaars. We prove that the map sends a Schubert class of the quantum K-theory ring of the flag variety to a closed K-k-Schur Katalan function up to an explicit factor related to a translation element with respect to an antidominant coroot. In fact, we prove this map coincides with a map whose existence was conjectured by Lam, Li, Mihalcea, Shimozono, and proved by Kato, and more recently by Chow and Leung. [ABSTRACT FROM AUTHOR]

Key-dependent message (KDM) security identity-based encryption (IBE) schemes aim to solve the security risks caused by the dependency between plaintext and secret keys in traditional IBE schemes. However, current KDM-IBE schemes are only secure with respect to affine functions, which limits their security level when a message is derived from the evaluation of a polynomial function using the secret key. To address this issue, in this study, we propose a novel approach to construct a KDM-IBE scheme with respect to polynomial or even arbitrary functions that achieves maximum security based on the learning with errors (LWE) assumption. Our approach overcomes two major technical barriers to constructing KDM-IBE schemes with respect to polynomial functions. Compared to existing KDM-IBE schemes, our proposed scheme ensures the secrecy of the key-related plaintext, even when it is obtained using arbitrary functions, not just affine functions. Thus, our approach provides a more robust solution to the security risks inherent in traditional IBE schemes. [ABSTRACT FROM AUTHOR]

We analyze the boundaries of the moduli spaces of compactifications of the heterotic string on Td, making particular emphasis on d = 2 and its F-theory dual. We compute the OPE algebras as we approach all the infinite distance limits that correspond to (possibly partial) decompactification limits in some dual frame. When decompactifying k directions, we find infinite towers of states becoming light that enhance the algebra arising at a given point in the moduli space of the Td−k compactification to its k-loop version, where the central extensions are given by the k KK vectors. For T2 compactifications, we reproduce all the affine algebras that arise in the F-theory dual, and show all the towers explicitly, including some that are not manifest in the F-theory counterparts. Furthermore, we construct the affine SO(32) algebra arising in the full decompactification limit, both in the heterotic and in the F-theory sides, showing that not only affine algebras of exceptional type arise in the latter. [ABSTRACT FROM AUTHOR]

Let G be a connected reductive group over a non-archimedean local field K, and assume that G splits over an unramified extension of K. We establish a local Langlands correspondence for irreducible unipotent representations of G. It comes as a bijection between the set of such representations (up to isomorphism) and the collection of enhanced L-parameters for G, which are trivial on the inertia subgroup of the Weil group of K. We show that this correspondence has many of the expected properties, for instance with respect to central characters, tempered representations, the discrete series, cuspidality and parabolic induction. The core of our strategy is the investigation of affine Hecke algebras on both sides of the local Langlands correspondence. When a Bernstein component of G-representations is matched with a Bernstein component of enhanced L-parameters, we prove a comparison theorem for the two associated affine Hecke algebras. This generalizes work of Lusztig in the case of adjoint K-groups. [ABSTRACT FROM AUTHOR]

Introduction: The current methodologies to quantify the palatal expansion are based on a preliminary rigid superimposition of 3D digital models representing the status of a given patient at different times. A new method based on affine alignment is proposed and compared to the gold standard, leading to the automatic analysis of 3-dimensional structural changes and to a simple numeric quantification of overall expansion vector and a better alignment of the digital models. Materials and methods: 40 digital models (timing span delta 25.8 ± 12.5 months) from young patients (mean age 10.7 ± 2.6) treated with two different palatal expansion techniques (20 subjects with RME—Rapid Maxillary Expander, and 20 subjects with NiTiSE, NiTi self-expander) were superimposed with the new affine alignment technique implemented as an extension package of the open-source MeshLab, from a golden standard starting point of rigid alignment. The results were then compared. Results: The new measurement function indicates a mean expansion expressed in a single numeric value of 9.3%, 10.3% for the RME group and 8.4% for the NiTiSE group respectively. The comparison with the golden standard showed a decrease to the average error from 0.91 mm to 0.58 mm. Conclusions: Affine alignment improves the current perspective of structural change quantification in the specific group of growing patients treated with palatal expanders giving the clinician useful information on the 3-dimensional morphological changes. [ABSTRACT FROM AUTHOR]

RGBT Salient Object Detection (SOD) focuses on common salient regions of a pair of visible and thermal infrared images. Existing methods perform on the well-aligned RGBT image pairs, but the captured image pairs are always unaligned and aligning them requires much labor cost. To handle this problem, we propose a novel deep correlation network (DCNet), which explores the correlations across RGB and thermal modalities, for weakly alignment-free RGBT SOD. In particular, DCNet includes a modality alignment module based on the spatial affine transformation, the feature-wise affine transformation and the dynamic convolution to model the strong correlation of two modalities. Moreover, we propose a novel bi-directional decoder model, which combines the coarse-to-fine and fine-to-coarse processes for better feature enhancement. In particular, we design a modality correlation ConvLSTM by adding the first two components of modality alignment module and a global context reinforcement module into ConvLSTM, which is used to decode hierarchical features in both top-down and button-up manners. Extensive experiments on three public benchmark datasets show the remarkable performance of our method against state-of-the-art methods. [ABSTRACT FROM AUTHOR]

We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive groups to the split reductive case and the pseudosplit pseudo-reductive commutative case. Moreover, we give the first results on the latter, including a rather complete description of the rank one case. [ABSTRACT FROM AUTHOR]

For a Dynkin quiver Q (of type \mathrm {ADE}), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc's monoidal category \mathcal {C}_{Q} of modules over the quantum loop algebra U_{q}(L\mathfrak {g}) via Nakajima's homomorphism. As an application, we show that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with Q and Hernandez-Leclerc's category \mathcal {C}_{Q}, assuming the simpleness of some poles of normalized R-matrices for type \mathrm {E}. [ABSTRACT FROM AUTHOR]

We consider groups defined by cyclic presentations where the defining word has length 3 and the cyclic presentation satisfies the T(6) small cancellation condition. We classify when these groups are hyperbolic. When combined with known results, this completely classifies the hyperbolic T(6) cyclically presented groups. [ABSTRACT FROM AUTHOR]

We prove the first direction of a recently posed conjecture by Gan, Gross and Prasad, which predicts branching laws that govern restriction from p-adic GLn to GLn-1 of irreducible smooth representations within the Arthur-type class. We extend this prediction to the full class of unitarizable representations, by exhibiting a combinatorial relation that must be satisfied for any pair of irreducible representations, in which one appears as a quotient of the restriction of the other. We settle the full conjecture for the cases in which either one of the representations in the pair is generic. The method of proof involves a transfer of the problem, using the Bernstein decomposition and the quantum affine Schur-Weyl duality, into the realm of quantum affine algebras. This restatement of the problem allows for an application of the combined power of a result of Hernandez on cyclic modules together with the Lapid-Mínguez criterion from the p-adic setting. [ABSTRACT FROM AUTHOR]