COMBINATORICS OF CANONICAL BASES REVISITED: STRING DATA IN TYPE A

We give a formula for the crystal structure on the integer points of the string polytopes and the *-crystal structure on the integer points of the string cones of type A for arbitrary reduced words. As a byproduct, we obtain defining inequalities for Nakashima–Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara *-involution on string data for a special choice of reduced word.

Analytic results for two-loop planar master integrals for Bhabha scattering

We analytically evaluate the master integrals for the second type of planar contributions to the massive two-loop Bhabha scattering in QED using differential equations with canonical bases. We obtain results in terms of multiple polylogarithms for all the master integrals but one, for which we derive a compact result in terms of elliptic multiple polylogarithms. As a byproduct, we also provide a compact analytic result in terms of elliptic multiple polylogarithms for an integral belonging to the first family of planar Bhabha integrals, whose computation in terms of polylogarithms was addressed previously in the literature.

Canonical bases arising from quantum symmetric pairs of Kac–Moody type

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$ -canonical bases for the highest weight integrable $\textbf U$ -modules and their tensor products regarded as $\textbf {U}^\imath$ -modules, as well as an $\imath$ -canonical basis for the modified form of the $\imath$ -quantum group $\textbf {U}^\imath$ . A key new ingredient is a family of explicit elements called $\imath$ -divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$ . We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi- $K$ -matrix and the constructions of $\imath$ -canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.

Influence of Non-canonical DNA Bases on the Genomic Diversity of Tevenvirinae

The Tevenvirinae viruses are some of the most common viruses on Earth. Representatives of this subfamily have long been used in the molecular biology studies as model organisms – since the emergence of the discipline. Tevenvirinae are promising agents for phage therapy in animals and humans, since their representatives have only lytic life cycle and many of their host bacteria are pathogens. As confirmed experimentally, some Tevenvirinae have non-canonical DNA bases. Non-canonical bases can play an essential role in the diversification of closely related viruses. The article performs a comparative and evolutionary analysis of Tevenvirinae genomes and components of Tevenvirinae genomes. A comparative analysis of these genomes and the genes associated with the synthesis of non-canonical bases allows us to conclude that non-canonical bases have a major influence on the divergence of Tevenvirinae viruses within the same habitats. Supposedly, Tevenvirinae developed a strategy for changing HGT frequency in individual populations, which was based on the accumulation of proteins for the synthesis of non-canonical bases and proteins that used those bases as substrates. Owing to this strategy, ancestors of Tevenvirinae with the highest frequency of HGT acquired genes that allowed them to exist in a certain niche, and ancestors with the lowest HGT frequency preserved the most adaptive of those genes. Given the origin and characteristics of genes associated with the synthesis of non-canonical bases in Tevenvirinae, one can assume that other phages may have similar strategies. The article demonstrates the dependence of genomic diversity of closely related Tevenvirinae on non-canonical bases.

Cones from quantum groups to tropical flag varieties

We relate quantum degree cones, parametrizing PBW degenerations of quantized enveloping algebras, to (negative tight monomial) cones introduced by Lusztig in the study of monomials in canonical bases, to K-theoretic cones for quiver representations, and to some maximal prime cones in tropical flag varieties.