Path combinatorics and light leaves for quiver Hecke algebras

We recast the classical notion of “standard tableaux" in an alcove-geometric setting and extend these classical ideas to all “reduced paths" in our geometry. This broader path-perspective is essential for implementing the higher categorical ideas of Elias–Williamson in the setting of quiver Hecke algebras. Our first main result is the construction of light leaves bases of quiver Hecke algebras. These bases are richer and encode more structural information than their classical counterparts, even in the case of the symmetric groups. Our second main result provides path-theoretic generators for the “Bott–Samelson truncation" of the quiver Hecke algebra.

A REALIZATION OF THE ENVELOPING SUPERALGEBRA

In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum $\mathfrak {gl}_n$ via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ of the loop algebra $\widehat {\mathfrak {gl}}_{m|n}$ of ${\mathfrak {gl}}_{m|n}$ with those of affine symmetric groups ${\widehat {{\mathfrak S}}_{r}}$ . Then, we give a BLM type realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ via affine Schur superalgebras. The first application of the realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ is to determine the action of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ on tensor spaces of the natural representation of $\widehat {\mathfrak {gl}}_{m|n}$ . These results in epimorphisms from $\;{\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ to affine Schur superalgebras so that the bridging relation between representations of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ and ${\widehat {{\mathfrak S}}_{r}}$ is established. As a second application, we construct a Kostant type $\mathbb Z$ -form for ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics.

Borel subalgebras of restricted Cartan-Type Lie algebras

It is still an open problem to determine the conjugacy classes of Borel subalgebras of non-classical type Lie algebras. In this paper, we prove that there are at least 2 conjugacy classes of Borel subalgebras as well as maximal triangulable subalgebras of restricted Cartan type Lie algebras of type W, S and H. We are particularly interested in maximal triangulable subalgebras of [Formula: see text] under some conditions which is called [Formula: see text]-subalgebras (Definition 3.1). We classify the conjugacy classes of [Formula: see text]-subalgebras for [Formula: see text] and determine their representatives. This paper and its sequel [Z. Lin, K. Ou and B. Shu, Geometric Setting of Jacobson–Witt Algebras, preprint] attempt to establish both algebraic and geometric setting for geometric representation theory of [Formula: see text]

Tropical optimal transport and Wasserstein distances

We study the problem of optimal transport in tropical geometry and define the Wasserstein-p distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric—a combinatorial metric that has been used to study of the tropical geometric space of phylogenetic trees—as the ground metric and study the cases of $$p=1,2$$ p = 1 , 2 in detail. The case of $$p=1$$ p = 1 gives an efficient computation of the infinitely-many geodesics on the tropical projective torus, while the case of $$p=2$$ p = 2 gives a form for Fréchet means and a general inner product structure. Our results also provide theoretical foundations for geometric insight a statistical framework in a tropical geometric setting. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport in tropical geometry. Several numerical examples are provided.

Finiteness Properties of Pseudo-Hyperbolic Varieties

Motivated by Lang–Vojta’s conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik’s theorem for dynamical systems of infinite order with properties of Prokhorov–Shramov’s notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again not only Amerik’s theorem and Prokhorov–Shramov’s notion of quasi-minimal model but also Weil’s regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of Kobayashi–Ochiai’s finiteness result for varieties of general type and thereby generalize Noguchi’s theorem (formerly Lang’s conjecture).

Hilbert’s 16th problem on a period annulus and Nash space of arcs

This paper introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert’s 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisor E of the canonical blow-up BI ℂn of the Bautin ideal I. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcs Arc(BI ℂn, E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.

A conditional limit theorem for high-dimensional ℓᵖ-spheres

The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.

Component-Centric Reduced Order Modeling for the Prediction of the Nonlinear Geometric Response of a Part of a Stiffened Structure

Component-centric reduced order models (ROMs) have recently been developed in the context of linear structural dynamics. They lead to an accurate prediction of the response of a part of structure (referred to as the β component) while not requiring a similar accuracy in the rest of the structure (referred to as the α component). The advantage of these ROMs over standard modal models is a significantly reduced number of generalized coordinates for structures with groups of close natural frequencies. This reduction is a very desirable feature for nonlinear geometric ROMs, and thus, the focus of the present investigation is on the formulation and validation of component-centric ROMs in the nonlinear geometric setting. The reduction in the number of generalized coordinates is achieved by rotating close frequency modes to achieve unobservable modes in the β component. In the linear case, these modes then completely disappear from the formulation owing to their orthogonality with the rest of the basis. In the nonlinear case, however, the generalized coordinates of these modes are still present in the nonlinear stiffness terms of the observable modes. A closure-type algorithm is then proposed to finally eliminate the unobserved generalized coordinates. This approach, its accuracy and computational savings, is demonstrated first on a simple beam model and then more completely on the 9-bay panel model considered in the linear investigation.