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k-Schur Functions and Affine Schubert Calculus

2014-06-05 Mathematics
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Release : 2014-06-05
Genre : Mathematics
Kind : eBook
Book Rating : 828/5 ( reviews)

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Book Synopsis k-Schur Functions and Affine Schubert Calculus by : Thomas Lam

Download or read book k-Schur Functions and Affine Schubert Calculus written by Thomas Lam. This book was released on 2014-06-05. Available in PDF, EPUB and Kindle. Book excerpt: 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

2014-06-30
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Release : 2014-06-30
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Kind : eBook
Book Rating : 833/5 ( reviews)

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Book Synopsis K-Schur Functions and Affine Schubert Calculus by : Thomas Lam

Download or read book K-Schur Functions and Affine Schubert Calculus written by Thomas Lam. This book was released on 2014-06-30. Available in PDF, EPUB and Kindle. Book excerpt:

The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions

2013-04-22 Mathematics
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Release : 2013-04-22
Genre : Mathematics
Kind : eBook
Book Rating : 94X/5 ( reviews)

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Book Synopsis The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions by : Thomas Lam

Download or read book The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions written by Thomas Lam. This book was released on 2013-04-22. Available in PDF, EPUB and Kindle. Book excerpt: 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

2020-10-24 Mathematics
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Release : 2020-10-24
Genre : Mathematics
Kind : eBook
Book Rating : 510/5 ( reviews)

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Book Synopsis Schubert Calculus and Its Applications in Combinatorics and Representation Theory by : Jianxun Hu

Download or read book Schubert Calculus and Its Applications in Combinatorics and Representation Theory written by Jianxun Hu. This book was released on 2020-10-24. Available in PDF, EPUB and Kindle. Book excerpt: 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

2010 Mathematics
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Release : 2010
Genre : Mathematics
Kind : eBook
Book Rating : 582/5 ( reviews)

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Book Synopsis Affine Insertion and Pieri Rules for the Affine Grassmannian by : Thomas Lam

Download or read book Affine Insertion and Pieri Rules for the Affine Grassmannian written by Thomas Lam. This book was released on 2010. Available in PDF, EPUB and Kindle. Book excerpt: 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.

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