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This book gives an introduction to the very active field of
combinatorics of affine Schubert calculus, explains the current
state of the art, and states the current open problems. Affine
Schubert calculus lies at the crossroads of combinatorics,
geometry, and representation theory. Its modern development is
motivated by two seemingly unrelated directions. One is the
introduction of k-Schur functions in the study of Macdonald
polynomial positivity, a mostly combinatorial branch of symmetric
function theory. The other direction is the study of the Schubert
bases of the (co)homology of the affine Grassmannian, an
algebro-topological formulation of a problem in enumerative
geometry. This is the first introductory text on this subject. It
contains many examples in Sage, a free open source general purpose
mathematical software system, to entice the reader to investigate
the open problems. This book is written for advanced undergraduate
and graduate students, as well as researchers, who want to become
familiar with this fascinating new field.
This book gives an introduction to the very active field of
combinatorics of affine Schubert calculus, explains the current
state of the art, and states the current open problems. Affine
Schubert calculus lies at the crossroads of combinatorics,
geometry, and representation theory. Its modern development is
motivated by two seemingly unrelated directions. One is the
introduction of k-Schur functions in the study of Macdonald
polynomial positivity, a mostly combinatorial branch of symmetric
function theory. The other direction is the study of the Schubert
bases of the (co)homology of the affine Grassmannian, an
algebro-topological formulation of a problem in enumerative
geometry. This is the first introductory text on this subject. It
contains many examples in Sage, a free open source general purpose
mathematical software system, to entice the reader to investigate
the open problems. This book is written for advanced undergraduate
and graduate students, as well as researchers, who want to become
familiar with this fascinating new field.
This unique book provides the first introduction to crystal base
theory from the combinatorial point of view. Crystal base theory
was developed by Kashiwara and Lusztig from the perspective of
quantum groups. Its power comes from the fact that it addresses
many questions in representation theory and mathematical physics by
combinatorial means. This book approaches the subject directly from
combinatorics, building crystals through local axioms (based on
ideas by Stembridge) and virtual crystals. It also emphasizes
parallels between the representation theory of the symmetric and
general linear groups and phenomena in combinatorics. The
combinatorial approach is linked to representation theory through
the analysis of Demazure crystals. The relationship of crystals to
tropical geometry is also explained.
This unique book provides the first introduction to crystal base
theory from the combinatorial point of view. Crystal base theory
was developed by Kashiwara and Lusztig from the perspective of
quantum groups. Its power comes from the fact that it addresses
many questions in representation theory and mathematical physics by
combinatorial means. This book approaches the subject directly from
combinatorics, building crystals through local axioms (based on
ideas by Stembridge) and virtual crystals. It also emphasizes
parallels between the representation theory of the symmetric and
general linear groups and phenomena in combinatorics. The
combinatorial approach is linked to representation theory through
the analysis of Demazure crystals. The relationship of crystals to
tropical geometry is also explained.
This is an introductory textbook designed for undergraduate
mathematics majors with an emphasis on abstraction and in
particular, the concept of proofs in the setting of linear algebra.
Typically such a student would have taken calculus, though the only
prerequisite is suitable mathematical grounding. The purpose of
this book is to bridge the gap between the more conceptual and
computational oriented undergraduate classes to the more abstract
oriented classes. The book begins with systems of linear equations
and complex numbers, then relates these to the abstract notion of
linear maps on finite-dimensional vector spaces, and covers
diagonalization, eigenspaces, determinants, and the Spectral
Theorem. Each chapter concludes with both proof-writing and
computational exercises.
This is an introductory textbook designed for undergraduate
mathematics majors with an emphasis on abstraction and in
particular, the concept of proofs in the setting of linear algebra.
Typically such a student would have taken calculus, though the only
prerequisite is suitable mathematical grounding. The purpose of
this book is to bridge the gap between the more conceptual and
computational oriented undergraduate classes to the more abstract
oriented classes. The book begins with systems of linear equations
and complex numbers, then relates these to the abstract notion of
linear maps on finite-dimensional vector spaces, and covers
diagonalization, eigenspaces, determinants, and the Spectral
Theorem. Each chapter concludes with both proof-writing and
computational exercises.
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