k-Schur Functions and Affine Schubert Calculus - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook k-Schur Functions and Affine Schubert Calculus write by Thomas Lam. This book was released on 2014-06-05. k-Schur Functions and Affine Schubert Calculus available in PDF, EPUB and Kindle. 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.
K-Schur Functions and Affine Schubert Calculus
K-Schur Functions and Affine Schubert Calculus - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook K-Schur Functions and Affine Schubert Calculus write by Thomas Lam. This book was released on 2014-06-30. K-Schur Functions and Affine Schubert Calculus available in PDF, EPUB and Kindle.
The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions
The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions write by Thomas Lam. This book was released on 2013-04-22. The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions available in PDF, EPUB and Kindle. The authors give a combinatorial expansion of a Schubert homology class in the affine Grassmannian $\mathrm{Gr}_{\mathrm{SL}_k}$ into Schubert homology classes in $\mathrm{Gr}_{\mathrm{SL}_{k+1}}$. This is achieved by studying the combinatorics of a new class of partitions called $k$-shapes, which interpolates between $k$-cores and $k+1$-cores. The authors define a symmetric function for each $k$-shape, and show that they expand positively in terms of dual $k$-Schur functions. They obtain an explicit combinatorial description of the expansion of an ungraded $k$-Schur function into $k+1$-Schur functions. As a corollary, they give a formula for the Schur expansion of an ungraded $k$-Schur function.
Schubert Calculus and Its Applications in Combinatorics and Representation Theory
Schubert Calculus and Its Applications in Combinatorics and Representation Theory - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Schubert Calculus and Its Applications in Combinatorics and Representation Theory write by Jianxun Hu. This book was released on 2020-10-24. Schubert Calculus and Its Applications in Combinatorics and Representation Theory available in PDF, EPUB and Kindle. This book gathers research papers and surveys on the latest advances in Schubert Calculus, presented at the International Festival in Schubert Calculus, held in Guangzhou, China on November 6–10, 2017. With roots in enumerative geometry and Hilbert's 15th problem, modern Schubert Calculus studies classical and quantum intersection rings on spaces with symmetries, such as flag manifolds. The presence of symmetries leads to particularly rich structures, and it connects Schubert Calculus to many branches of mathematics, including algebraic geometry, combinatorics, representation theory, and theoretical physics. For instance, the study of the quantum cohomology ring of a Grassmann manifold combines all these areas in an organic way. The book is useful for researchers and graduate students interested in Schubert Calculus, and more generally in the study of flag manifolds in relation to algebraic geometry, combinatorics, representation theory and mathematical physics.
Affine Insertion and Pieri Rules for the Affine Grassmannian
Affine Insertion and Pieri Rules for the Affine Grassmannian - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Affine Insertion and Pieri Rules for the Affine Grassmannian write by Thomas Lam. This book was released on 2010. Affine Insertion and Pieri Rules for the Affine Grassmannian available in PDF, EPUB and Kindle. The authors study combinatorial aspects of the Schubert calculus of the affine Grassmannian ${\rm Gr}$ associated with $SL(n,\mathbb{C})$.Their main results are: Pieri rules for the Schubert bases of $H^*({\rm Gr})$ and $H_*({\rm Gr})$, which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. A new combinatorial definition for $k$-Schur functions, which represent the Schubert basis of $H_*({\rm Gr})$. A combinatorial interpretation of the pairing $H^*({\rm Gr})\times H_*({\rm Gr}) \rightarrow\mathbb Z$ induced by the cap product.
