Affine Insertion and Pieri Rules for the Affine Grassmannian

Affine Insertion and Pieri Rules for the Affine Grassmannian

Author: Thomas Lam

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 103

ISBN-13: 0821846582

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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.


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 and published by American Mathematical Soc.. This book was released on 2010 with total page 103 pages. 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.


Affine Insertion and Pieri Rules for the Affine Grassmannian

Affine Insertion and Pieri Rules for the Affine Grassmannian

Author: Thomas Lam

Publisher:

Published: 2010

Total Pages: 82

ISBN-13: 9781470405915

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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 and published by . This book was released on 2010 with total page 82 pages. Available in PDF, EPUB and Kindle. Book excerpt:


Affine Insertion and Pieri Rules for the Affine Grassmannian

Affine Insertion and Pieri Rules for the Affine Grassmannian

Author:

Publisher: American Mathematical Soc.

Published:

Total Pages: 103

ISBN-13: 0821867180

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"Volume 208, number 977 (second of 6 numbers)."


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

Download or read book Affine Insertion and Pieri Rules for the Affine Grassmannian written by and published by American Mathematical Soc.. This book was released on with total page 103 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Volume 208, number 977 (second of 6 numbers)."


k-Schur Functions and Affine Schubert Calculus

k-Schur Functions and Affine Schubert Calculus

Author: Thomas Lam

Publisher: Springer

Published: 2014-06-05

Total Pages: 226

ISBN-13: 1493906828

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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.


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 and published by Springer. This book was released on 2014-06-05 with total page 226 pages. 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.


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

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

Author: Thomas Lam

Publisher: American Mathematical Soc.

Published: 2013-04-22

Total Pages: 113

ISBN-13: 082187294X

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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.


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 and published by American Mathematical Soc.. This book was released on 2013-04-22 with total page 113 pages. 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.


Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics

Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics

Author: Mahir Can

Publisher: Springer

Published: 2014-06-11

Total Pages: 360

ISBN-13: 149390938X

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This book contains a collection of fifteen articles and is dedicated to the sixtieth birthdays of Lex Renner and Mohan Putcha, the pioneers of the field of algebraic monoids. Topics presented include: structure and representation theory of reductive algebraic monoids monoid schemes and applications of monoids monoids related to Lie theory equivariant embeddings of algebraic groups constructions and properties of monoids from algebraic combinatorics endomorphism monoids induced from vector bundles Hodge–Newton decompositions of reductive monoids A portion of these articles are designed to serve as a self-contained introduction to these topics, while the remaining contributions are research articles containing previously unpublished results, which are sure to become very influential for future work. Among these, for example, the important recent work of Michel Brion and Lex Renner showing that the algebraic semi groups are strongly π-regular. Graduate students as well as researchers working in the fields of algebraic (semi)group theory, algebraic combinatorics and the theory of algebraic group embeddings will benefit from this unique and broad compilation of some fundamental results in (semi)group theory, algebraic group embeddings and algebraic combinatorics merged under the umbrella of algebraic monoids.


Book Synopsis Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics by : Mahir Can

Download or read book Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics written by Mahir Can and published by Springer. This book was released on 2014-06-11 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains a collection of fifteen articles and is dedicated to the sixtieth birthdays of Lex Renner and Mohan Putcha, the pioneers of the field of algebraic monoids. Topics presented include: structure and representation theory of reductive algebraic monoids monoid schemes and applications of monoids monoids related to Lie theory equivariant embeddings of algebraic groups constructions and properties of monoids from algebraic combinatorics endomorphism monoids induced from vector bundles Hodge–Newton decompositions of reductive monoids A portion of these articles are designed to serve as a self-contained introduction to these topics, while the remaining contributions are research articles containing previously unpublished results, which are sure to become very influential for future work. Among these, for example, the important recent work of Michel Brion and Lex Renner showing that the algebraic semi groups are strongly π-regular. Graduate students as well as researchers working in the fields of algebraic (semi)group theory, algebraic combinatorics and the theory of algebraic group embeddings will benefit from this unique and broad compilation of some fundamental results in (semi)group theory, algebraic group embeddings and algebraic combinatorics merged under the umbrella of algebraic monoids.


Finite Order Automorphisms and Real Forms of Affine Kac-Moody Algebras in the Smooth and Algebraic Category

Finite Order Automorphisms and Real Forms of Affine Kac-Moody Algebras in the Smooth and Algebraic Category

Author: Ernst Heintze

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 81

ISBN-13: 0821869183

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Heintze and Gross discuss isomorphisms between smooth loop algebras and of smooth affine Kac-Moody algebras in particular, and automorphisms of the first and second kinds of finite order. Then they consider involutions of the first and second kind, and make the algebraic case. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com).


Book Synopsis Finite Order Automorphisms and Real Forms of Affine Kac-Moody Algebras in the Smooth and Algebraic Category by : Ernst Heintze

Download or read book Finite Order Automorphisms and Real Forms of Affine Kac-Moody Algebras in the Smooth and Algebraic Category written by Ernst Heintze and published by American Mathematical Soc.. This book was released on 2012 with total page 81 pages. Available in PDF, EPUB and Kindle. Book excerpt: Heintze and Gross discuss isomorphisms between smooth loop algebras and of smooth affine Kac-Moody algebras in particular, and automorphisms of the first and second kinds of finite order. Then they consider involutions of the first and second kind, and make the algebraic case. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com).


Recent Trends in Algebraic Combinatorics

Recent Trends in Algebraic Combinatorics

Author: Hélène Barcelo

Publisher: Springer

Published: 2019-01-21

Total Pages: 362

ISBN-13: 3030051412

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This edited volume features a curated selection of research in algebraic combinatorics that explores the boundaries of current knowledge in the field. Focusing on topics experiencing broad interest and rapid growth, invited contributors offer survey articles on representation theory, symmetric functions, invariant theory, and the combinatorics of Young tableaux. The volume also addresses subjects at the intersection of algebra, combinatorics, and geometry, including the study of polytopes, lattice points, hyperplane arrangements, crystal graphs, and Grassmannians. All surveys are written at an introductory level that emphasizes recent developments and open problems. An interactive tutorial on Schubert Calculus emphasizes the geometric and topological aspects of the topic and is suitable for combinatorialists as well as geometrically minded researchers seeking to gain familiarity with relevant combinatorial tools. Featured authors include prominent women in the field known for their exceptional writing of deep mathematics in an accessible manner. Each article in this volume was reviewed independently by two referees. The volume is suitable for graduate students and researchers interested in algebraic combinatorics.


Book Synopsis Recent Trends in Algebraic Combinatorics by : Hélène Barcelo

Download or read book Recent Trends in Algebraic Combinatorics written by Hélène Barcelo and published by Springer. This book was released on 2019-01-21 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: This edited volume features a curated selection of research in algebraic combinatorics that explores the boundaries of current knowledge in the field. Focusing on topics experiencing broad interest and rapid growth, invited contributors offer survey articles on representation theory, symmetric functions, invariant theory, and the combinatorics of Young tableaux. The volume also addresses subjects at the intersection of algebra, combinatorics, and geometry, including the study of polytopes, lattice points, hyperplane arrangements, crystal graphs, and Grassmannians. All surveys are written at an introductory level that emphasizes recent developments and open problems. An interactive tutorial on Schubert Calculus emphasizes the geometric and topological aspects of the topic and is suitable for combinatorialists as well as geometrically minded researchers seeking to gain familiarity with relevant combinatorial tools. Featured authors include prominent women in the field known for their exceptional writing of deep mathematics in an accessible manner. Each article in this volume was reviewed independently by two referees. The volume is suitable for graduate students and researchers interested in algebraic combinatorics.


Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions

Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions

Author: Tom H. Koornwinder

Publisher: Cambridge University Press

Published: 2020-10-15

Total Pages: 442

ISBN-13: 1108916554

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This is the second of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 2 covers multivariable special functions. When the Bateman project appeared, study of these was in an early stage, but revolutionary developments began to be made in the 1980s and have continued ever since. World-renowned experts survey these over the course of 12 chapters, each containing an extensive bibliography. The reader encounters different perspectives on a wide range of topics, from Dunkl theory, to Macdonald theory, to the various deep generalizations of classical hypergeometric functions to the several variables case, including the elliptic level. Particular attention is paid to the close relation of the subject with Lie theory, geometry, mathematical physics and combinatorics.


Book Synopsis Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions by : Tom H. Koornwinder

Download or read book Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions written by Tom H. Koornwinder and published by Cambridge University Press. This book was released on 2020-10-15 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the second of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 2 covers multivariable special functions. When the Bateman project appeared, study of these was in an early stage, but revolutionary developments began to be made in the 1980s and have continued ever since. World-renowned experts survey these over the course of 12 chapters, each containing an extensive bibliography. The reader encounters different perspectives on a wide range of topics, from Dunkl theory, to Macdonald theory, to the various deep generalizations of classical hypergeometric functions to the several variables case, including the elliptic level. Particular attention is paid to the close relation of the subject with Lie theory, geometry, mathematical physics and combinatorics.


Robin Functions for Complex Manifolds and Applications

Robin Functions for Complex Manifolds and Applications

Author: Kang-Tae Kim

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 126

ISBN-13: 0821849654

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"Volume 209, number 984 (third of 5 numbers)."


Book Synopsis Robin Functions for Complex Manifolds and Applications by : Kang-Tae Kim

Download or read book Robin Functions for Complex Manifolds and Applications written by Kang-Tae Kim and published by American Mathematical Soc.. This book was released on 2011 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Volume 209, number 984 (third of 5 numbers)."