This handbook presents a thorough introduction to current topics of mathematical research in combinatorial algebraic geometry. The editors’ aim is to introduce researchers to key literature from the past 20-30 years needed to address open questions in the field. The chapters give concrete, computational examples of Lie-theoretic and combinatorial tools applied to the geometry of flag varieties and their subvarieties.

Lie theory provides a common language for the articles in this text, so while chapters are self-contained, it is recommended readers have some prior familiarity with the foundations of the subject. Each chapter benefits multiple sets of readers including:

• Graduate students seeking to conduct research in algebraic combinatorics and Lie theory. New researchers will be introduced to relevant techniques used to prove key results and gain insight from leading researchers into the context of these results.
• Experts in the field seeking insights and exposure to techniques and the finer expository points of related topics.
• Mathematicians looking for a centralized reference on the geometry and combinatorics of flag varieties.

The topics of this handbook break down into four sections. The first section of this book consists of an introduction to the cohomology of flag varieties, Schubert varieties, and Schubert polynomials. The second section explores subvarieties of the flag variety that generalize or complement Schubert varieties in various ways. The third section of the book focuses on Hessenberg varieties. Finally, the last section explores additional topics related to flag varieties.

Last, the editors include a brief word about a few things this book does not do. Although great care is taken to streamline notation, the avid reader will still find variation throughout the chapters. This is reflective of, and prepares the reader for, the state of the field. For example, different notations for Richardson varieties typically appear in work on positivity than in other subfields of combinatorial algebraic geometry.

The editors and contributors hope readers find this book useful and enjoyable.