A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces

An explicit determination of the $K$-theoretic structure constants of the affine Grassmannian associated to $SL_2$

Let $G:=\widehat{SL_2}$ denote the affine Kac-Moody group associated to $SL_2$ and $\bar{\mathcal{X}}$ the associated affine Grassmannian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant Grothendieck group of $\bar{\mathcal{X}}$. In the case of ordinary (non-equivariant) $K$-theory we find an explicit closed form for the structure constants. We also determine an inductive formula for the structure constants in the torus-equivariant cohomology ring, and use this formula to find closed forms for some of the structure constants.

Pieri rules for the K-theory of cominuscule Grassmannians

We prove Pieri formulas for the multiplication with special Schubert classes in the K-theory of all cominuscule Grassmannians. For Grassmannians of type A this gives a new proof of a formula of Lenart. Our formula is new for Lagrangian Grassmannians, and for orthogonal Grassmannians it proves a special case of a conjectural Littlewood-Richardson rule of Thomas and Yong. Recent work of Thomas and Yong and of E. Clifford has shown that the full Littlewood-Richardson rule for orthogonal Grassmannians follows from the Pieri case proved here. We describe the K-theoretic Pieri coefficients both as i…

仿射格拉斯曼流形的等变 $K$-同调与 $K$-理论双重 $k$-Schur 函数

Equivariant $K$-homology of affine Grassmannian and $K$-theoretic double $k$-Schur functions

本文研究仿射格拉斯曼流形的环面等变 $K$-同调环，在 $A$ 型中引入 $K$-理论双重 $k$-Schur 函数作为 Schubert 基，并给出 $\mathrm{Gr}_{{SL}_n}$ 的等变 $K$-同调环的 Ginzburg-Peterson 型实现。

K-theory of minuscule varieties

Based on Thomas and Yong's K-theoretic jeu de taquin algorithm, we prove a uniform Littlewood-Richardson rule for the K-theoretic Schubert structure constants of all minuscule homogeneous spaces. Our formula is new in all types. For the main examples of Grassmannians of type A and maximal orthogonal Grassmannians it has the advantage that the tableaux to be counted can be recognized without reference to the jeu de taquin algorithm.

A Lower Bound on the Density of Sphere Packings via Graph Theory

Using graph-theoretic methods we give a new proof that for all sufficiently large $n$, there exist sphere packings in $\R^n$ of density at least $cn2^{-n}$, exceeding the classical Minkowski bound by a factor linear in $n$. This matches up to a constant the best known lower bounds on the density of sphere packings due to Rogers, Davenport-Rogers, and Ball. The suggested method makes it possible to describe the points of such a packing with complexity $\exp(n\log n)$, which is significantly lower than in the other approaches.

Note on a Conjecture of Wegner

The optimal packings of n unit discs in the plane are known for those natural numbers n, which satisfy certain number theoretic conditions. Their geometric realizations are the extremal Groemer packings (or Wegner packings). But an extremal Groemer packing of n unit discs does not exist for all natural numbers n and in this case, the number n is called exceptional. We are interested in number theoretic characterizations of the exceptional numbers. A counterexample is given to a conjecture of Wegner concerning such a characterization. We further give a characterization of the exceptional number…

A representation-theoretic interpretation of positroid classes

A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology class Poincaré dual to a positroid variety. We define a family of representations of general linear groups whose characters are symmetric polynomials representing positroid classes. These representations are certain diagram Schur modules in the sense of James and Peel. This gives a new algebraic interpretation of the Schubert structure constants for the product…

从格拉斯曼流形到单纯高维扩张器

From Grassmannian to Simplicial High-Dimensional Expanders

本文提出一种基于格拉斯曼HDX的凯莱单纯复形构造，实现次多项式度数与任意好局部谱扩张，并证明其1-同调群维数可达Ω(log²N)，超越以往构造。

任意范数下球体堆积的下界

Lower bounds for sphere packing in arbitrary norms

本文证明任意d维实赋范空间中单位球堆积密度下界为(1-o(1))d log d/2^(d+1)，改进Schmidt结果并推广至一般范数，方法结合图论与凸几何体积界。

图的香农容量的群论方法与格填充的极限定理

A group-theoretic approach to Shannon capacity of graphs and a limit theorem from lattice packings

本研究提出香农容量问题的群论方法，统一推广已有下界，强化环图极限结论至所有分数图，构造格结构独立集并规避了已知的线性构造障碍。

装箱问题的最大-最小与1-有界空间算法

Max-Min and 1-Bounded Space Algorithms for the Bin Packing Problem

本文研究装箱问题中最大-最小与1-有界空间两类算法的性能，证明MM算法渐近近似比至多1.5，给出两类算法交集的下界1.25，并将分析扩展到基数约束装箱问题。

闭 $k$-Schur Katalan 函数作为仿射 Grassmann 流形 $K$-同调中的 Schubert 代表

Closed $k$-Schur Katalan functions as $K$-homology Schubert representatives of the affine Grassmannian

本文证明闭 $k$-Schur Katalan 函数是仿射 Grassmann 流形 $K$-同调的 Schubert 代表，并阐明其与量子 $K$-理论 Peterson 同构的关系。