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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.

The Poset of [kappa]-shapes and Branching Rules for [kappa]-Schur Functions

2013 Partially ordered sets
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Release : 2013
Genre : Partially ordered sets
Kind : eBook
Book Rating : 741/5 ( reviews)

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Book Synopsis The Poset of [kappa]-shapes and Branching Rules for [kappa]-Schur Functions by : Thomas Lam

Download or read book The Poset of [kappa]-shapes and Branching Rules for [kappa]-Schur Functions written by Thomas Lam. This book was released on 2013. Available in PDF, EPUB and Kindle. Book excerpt: We give a combinatorial expansion of a Schubert homology class in the affine Grassmannian GrSLk into Schubert homology classes in GrSLk+1. This is achieved by studying the combinatorics of a new class of partitions called k-shapes, which interpolates between k k-cores and kk+1-cores. We define a symmetric function for each k-shape, and show that they expand positively in terms of dual k-Schur functions. We obtain an explicit combinatorial description of the expansion of an ungraded k k-Schur function into k+1-Schur functions. As a corollary, we give a formula for the Schur expansion of an ungraded k-Schur function.

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.

On Central Critical Values of the Degree Four $L$-Functions for GSp(4): The Fundamental Lemma. III

2013-08-23 Mathematics
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Release : 2013-08-23
Genre : Mathematics
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Book Rating : 424/5 ( reviews)

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Book Synopsis On Central Critical Values of the Degree Four $L$-Functions for GSp(4): The Fundamental Lemma. III by : Masaaki Furusawa

Download or read book On Central Critical Values of the Degree Four $L$-Functions for GSp(4): The Fundamental Lemma. III written by Masaaki Furusawa. This book was released on 2013-08-23. Available in PDF, EPUB and Kindle. Book excerpt: Some time ago, the first and third authors proposed two relative trace formulas to prove generalizations of Böcherer's conjecture on the central critical values of the degree four -functions for , and proved the relevant fundamental lemmas. Recently, the first and second authors proposed an alternative third relative trace formula to approach the same problem and proved the relevant fundamental lemma. In this paper the authors extend the latter fundamental lemma and the first of the former fundamental lemmas to the full Hecke algebra. The fundamental lemma is an equality of two local relative orbital integrals. In order to show that they are equal, the authors compute them explicitly for certain bases of the Hecke algebra and deduce the matching.

Non-cooperative Equilibria of Fermi Systems with Long Range Interactions

2013-06-28 Mathematics
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Release : 2013-06-28
Genre : Mathematics
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Book Rating : 761/5 ( reviews)

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Book Synopsis Non-cooperative Equilibria of Fermi Systems with Long Range Interactions by : Jean-Bernard Bru

Download or read book Non-cooperative Equilibria of Fermi Systems with Long Range Interactions written by Jean-Bernard Bru. This book was released on 2013-06-28. Available in PDF, EPUB and Kindle. Book excerpt: The authors define a Banach space $\mathcal{M}_{1}$ of models for fermions or quantum spins in the lattice with long range interactions and make explicit the structure of (generalized) equilibrium states for any $\mathfrak{m}\in \mathcal{M}_{1}$. In particular, the authors give a first answer to an old open problem in mathematical physics--first addressed by Ginibre in 1968 within a different context--about the validity of the so-called Bogoliubov approximation on the level of states. Depending on the model $\mathfrak{m}\in \mathcal{M}_{1}$, the authors' method provides a systematic way to study all its correlation functions at equilibrium and can thus be used to analyze the physics of long range interactions. Furthermore, the authors show that the thermodynamics of long range models $\mathfrak{m}\in \mathcal{M}_{1}$ is governed by the non-cooperative equilibria of a zero-sum game, called here thermodynamic game.

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