Algebraic combinatorics

Algebraic combinatorics is a branch of mathematics that combines techniques from algebra, specifically linear algebra and abstract algebra, with combinatorial methods to solve problems related to discrete structures, counting, and arrangements. This area of study often involves the interplay between combinatorial objects (like graphs, permutations, and sets) and algebraic structures (like groups, rings, and fields).

Buekenhout geometry is a type of combinatorial geometry that involves the study of certain kinds of incidence structures called "generalized polygons." Specifically, it is named after the mathematician F. Buekenhout, who contributed significantly to the field of incidence geometry.

In the context of mathematics, particularly in combinatorial geometry and geometric combinatorics, a "building" refers to a particular type of geometric structure that generalizes the concept of certain types of spaces, often associated with groups of symmetries known as "Lie groups." Buildings are combinatorial structures that can be used to study these groups and their representations. Buildings can be defined as a collection of simplices that meet specific conditions, producing a coherent geometric structure.

Combinatorial commutative algebra is a branch of mathematics that merges concepts from commutative algebra with combinatorial techniques and ideas. This field studies algebraic objects (like ideals, rings, and varieties) using combinatorial methods, often involving graph theory, polytopes, and combinatorial configurations.

Combinatorial species is a concept from combinatorics and algebraic combinatorics that provides a framework for studying and enumerating combinatorial structures through the use of the theory of functors. The notion of species was developed primarily by André Joyal in the 1980s to capture and formalize the combinatorial properties of various structures.

"Combinatorics: The Rota Way" is a book authored by Richard P. Stanley, which aims to explore the field of combinatorics through the lens of the influential mathematician Gian-Carlo Rota. The book emphasizes Rota's insights and perspectives, particularly regarding enumerative combinatorics, posets (partially ordered sets), and various combinatorial structures.

The Coxeter complex is a mathematical concept that arises in the field of geometry and combinatorial group theory. It is closely associated with Coxeter groups, which are groups generated by reflections across hyperplanes in a Euclidean space. The Coxeter complex provides a way to visualize the geometric structure related to these groups.

A **differential poset** (short for "differential partially ordered set") is a concept used in the study of combinatorics and order theory. While the term itself is not universally defined across all areas of mathematics, it generally refers to a partially ordered set (poset) that has some structure or properties related to differential operations, which might be in the context of algebraic structures or certain combinatorial interpretations.

Dominance order is a concept used in various fields, including economics, game theory, and biology, to describe a hierarchical relationship where one element is more dominant or superior compared to another. Here are a few contexts in which dominance order is commonly applied: 1. **Game Theory**: In game theory, dominance order refers to strategies that are superior to others regardless of what opponents choose. A dominant strategy is one that results in a better payoff for a player, regardless of what the other players do.

An Eulerian poset, or Eulerian partially ordered set, is a type of partially ordered set (poset) that satisfies certain combinatorial properties related to its rank.

A **finite ring** is a ring that contains a finite number of elements. In more formal terms, a ring \( R \) is an algebraic structure consisting of a set equipped with two binary operations, typically referred to as addition and multiplication, that satisfy certain properties: 1. **Addition**: - \( R \) is an abelian group under addition. This means that: - There exists an additive identity (usually denoted as \( 0 \)).

Garnir relations refer to a specific set of algebraic identities that arise in the context of representation theory and the study of certain mathematical structures, particularly in relation to symmetric groups and permutation representations. Named after the mathematician Jean Garnir, these relations are particularly important in the study of the modular representation theory of symmetric groups and their related structures.

A **graded poset** (partially ordered set) is a specific type of poset that has an additional structure related to its elements' ranks or levels. Here are the key characteristics of a graded poset: 1. **Partially Ordered Set**: A graded poset is first and foremost a poset, meaning it consists of a set of elements paired with a binary relation that is reflexive, antisymmetric, and transitive.

An H-vector is a concept that arises in the context of algebraic topology and combinatorial structures, particularly in the study of partially ordered sets (posets) and their associated simplicial complexes. The H-vector is often related to the notation used for the generating function of a simplicial complex or the f-vector of a polytope.

A Hessenberg variety is a type of algebraic variety that arises in the context of representations of Lie algebras and algebraic geometry. Specifically, Hessenberg varieties are associated with a choice of a nilpotent operator on a vector space and a subspace that captures certain "Hessenberg" conditions. They can be thought of as a geometric way to study certain types of matrices or linear transformations up to a specified degree of nilpotency.

Incidence algebra is a branch of algebra that deals with the study of incidence relations among a set of objects, usually within the context of partially ordered sets (posets) or other combinatorial structures. The main aim is to analyze and represent relationships between elements in these structures through algebraic constructs. In a typical incidence algebra, one often considers a poset \( P \) and defines an algebraic structure where the elements are functions defined on the pairs of elements in the poset.

The Kruskal-Katona theorem is a result in combinatorial set theory, particularly related to the theory of hypergraphs and the study of families of sets. It provides a connection between the structure of a family of sets and the number of its intersections. The theorem defines conditions under which an antipodal family (a family of subsets) can be characterized in terms of its lower shadow, which is a fundamental concept in combinatorics.

A lattice word is a concept primarily used in the fields of combinatorics and formal language theory. It refers to a specific arrangement of symbols that can be visualized as a word in a lattice structure. In more technical terms, a lattice word typically arises when considering combinatorial objects associated with lattice paths. In a combinatorial context, a common interpretation of lattice words involves considering strings that correspond to paths on a grid.

The Littelmann path model is a combinatorial framework used to study representations of semisimple Lie algebras and their quantum analogs. Introduced by Philip Littelmann in the mid-1990s, this model provides a geometric interpretation of the representation theory through the use of paths in a certain combinatorial structure.

Restricted representation, in various contexts, generally refers to a method or framework that limits or confines the scope of representation in some way. The exact meaning can vary depending on the field of study or application: 1. **Mathematics and Abstract Algebra**: In this context, restricted representation often refers to representations of algebraic structures (like groups or algebras) that are limited to a certain subset of their elements.

In algebraic geometry, a Schubert variety is a particular type of subvariety of a flag variety, which in turn is a parameter space for certain types of subspaces of a vector space. Schubert varieties arise in the study of intersection theory, representation theory, and several other areas of mathematics.

A simplicial sphere is a type of topological space that arises in the field of algebraic topology and combinatorial geometry. More specifically, it is a simplicial complex that is homeomorphic to a sphere. ### Definition A **simplicial complex** is a set of simplices that satisfies certain conditions, such as closure under taking faces and the intersection property.

Stanley's reciprocity theorem, named after mathematician Richard P. Stanley, is a result in combinatorial mathematics, particularly in the field of algebraic combinatorics and the study of combinatorial structures such as generating functions and posets (partially ordered sets). The theorem relates to the generating functions of certain combinatorial structures, specifically in the context of the polynomial ring and symmetric functions.

A Stanley–Reisner ring, also known as a face ring or a simplicial ring, is a particular type of graded ring that is associated with a simplicial complex. The construction of a Stanley–Reisner ring arises in the field of combinatorial commutative algebra and algebraic geometry, especially in the study of toric varieties and posets.