Discrete mathematics 72 1988 385390 385 northholland on the littlewood richardson rule in terms of lattice path combinatorics toshihiro watanabe department of applied mathematics, faculty of engineering, gifu university, yanagido 11, gifu, 50111, japan received 22 august 1986 revised 9 april 1987 this work presents a simple proof of the littlewood richardson rule on multiplying schur. A geometric littlewood richardson rule ravi vakil stanford university abstract littlewood richardson coefficients are fundamental constants in several fields of mathematics and in nature. An introduction is given to the littlewood richardson rule, and various combinatorial constructions related to it. Math 99r, algebraic combinatorics symmetric functions possible project topics the nal project should be a 710 page expository paper on topic related to symmetric functions. The key ingredient in the proof is the geometric littlewood richardson rule, described in a companion paper. The archetypal example of such a formula is the littlewoodrichardson rule, which gives the multiplicities. Combinatorics and geometry of littlewoodrichardson. We consider this question because the related algebraic combinatorics are particularly elegant, and hence have expository value in connecting harmonic analysis with algebraic combinatorics. Read pdf enumerative combinatorics volume 2 for full. In particular, we prove a version of the littlewood richardson rule for skew schur functions that was conjectured by assaf and mcnamara am, and which extends their skew pieri formula.
An introduction to symmetric functions and their combinatorics. A littlewood richardson rule for the ktheory of grassmannians 39 this bialgebra f can be seen as a ktheory parallel of the ring of symmetric functions 20, 10, which in a similar way describes the cohomology of grassmannians, in addition. A littlewoodrichardson rule for macdonald polynomials. Littlewoodrichardson rules, such as tableaux, and knutson and taos puzzles. On the structures of hive algebras and tensor product. Much of algebraic combinatorics is motivated by the search for positive combinatorial formulae for quantities that are known to be nonnegative by geometry or representation theory 29. Also covered are connections between symmetric functions and representation theory. An introduction is given to the littlewoodrichardson rule, and various combinatorial constructions related to it. Combinatorics and geometry of littlewoodrichardson cones. Parts of it are closely related to representation theory and commutative algebra. Skew littlewoodrichardson rules from hopf algebras thomas lam, aaron lauve, and frank sottile abstract. A littlewoodrichardson rule for grassmannian permutations article pdf available in proceedings of the american mathematical society 76 september 2007 with 41 reads how we measure reads.
A geometric littlewood richardson rule by ravi vakil abstract we describe a geometric littlewood richardson rule, interpreted as deforming the intersection of two schubert varieties into the union of schubert varieties. In the past decade the littlewoodrichardson rule lr rule has moved into. Intheprequeltothispaper,weshowedhowresultsofmasoninvolving a new combinatorial formula for polynomials that are now known as demazure atoms characters of quotients of demazure modules. We use hopf algebras to prove a version of the littlewoodrichardson rule for skew schur functions, which implies a conjecture of assaf and mcnamara. On the onehand it can be thought of as describing induction in symmetric groups. We conclude with a section relating these fairly modern techniques to earlier work on the littlewood richardson rule. Mar 28, 2012 macdonald polynomials are orthogonal polynomials associated to root systems, and in the type a case, the symmetric macdonald polynomials are a common generalization of schur functions, macdonald spherical functions, and jack polynomials. On the littlewoodrichardson rule in terms of lattice path. We conclude with a section relating these fairly modern techniques to earlier work on.
S 01 article electronically published on october 12, enumerative combinatorics, volume 2, by richard p. Vallejo european journal of combinatorics 26 2005 9951008 fig. Much of algebraic combinatorics is motivated by the search for positive combinatorial formulae for quantities that are known to be nonnegative by geometry or representation theory 49. Applications and combinatorics in algebraic geometry. In mathematics, the littlewoodrichardson rule is a combinatorial description of the.
We present a proof based on tableau switching, dual equivalence, and coplactic operations. Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type a case, the symmetric macdonald polynomials are a common generalization of schur functions, macdonald spherical functions, and jack polynomials. The key ingredient in the proof is the geometric littlewoodrichardson rule, described in a companion paper. The archetypal example of such a formula is the littlewood richardson rule, which gives the multiplicities.
A geometric littlewoodrichardson rule ravi vakil stanford university abstract littlewoodrichardson coefficients are fundamental constants in several fields of mathematics and in nature. We also establish skew littlewood richardson rules for schur p and qfunctions and noncommu. The littlewoodrichardson rule is a combinatorial rule describing the multiplication of schur polynomials. As in volume 1, the exercises play a vital role in developing the material. These coefficients are natural numbers, which the littlewoodrichardson rule describes as counting certain skew tableaux. Define the bottsamelson variety bsq associated to a quilt q to be the. The littlewoodrichardson rule is a fundamental formula in combinatorics and representation theory. Summer reus research experiences for undergrads vic reiner i started mentoring reus in summer of 2000, and since 2005 have been comentoring them with various of my combinatorial and algebraic colleagues, christine berkesch, ben brubaker, gregg musiker, pavlo pylyavskyy, dennis stanton. Classical applications to representation theory of the symmetric and the full linear group, to the symmetric functions, etc. They have more general lietheoretic analogs weyls characters, etc.
A littlewoodrichardson rule for the k theory of grassmannians. A littlewoodrichardson rule is a combinatorial interpretation of the littlewoodrichardson. Its combinatorial approach is readerfriendly and invites the reader to get involved, either working on the exercises listed at the end of each chapter or by playing with the computational implementation on freelyavailable software, sage in this case, where the authors have many builtin. We use hopf algebras to prove a version of the littlewood richardson rule for skew schur functions, which implies a conjecture of assaf and mcnamara.
A concise proof of the littlewoodrichardson rule emis the. Know that ebook versions of most of our titles are still available and may be downloaded immediately after purchase. In the past decade the littlewoodrichardson rule lr rule moved into a. It provides an explicit combinatorial rule for expressing either a. Skew littlewood richardson rules from hopf algebras thomas lam, aaron lauve, and frank sottile abstract. The odd littlewoodrichardson rule article pdf available in journal of algebraic combinatorics 374 november 2011 with reads how we measure reads. Interaction of combinatorics and representation theory, 95145, the mathematical society of japan, tokyo, japan, 2001. The introduction to crystal bases given in this book is accessible to graduate students and researchers. The second is more combinatorial, involving permutation. The archetypal example of such a formula is the littlewood richardson rule, which gives the mul. The littlewoodrichardson rule is one of the most important results in the theory of symmetric functions. The result is a direct combinatorial interpretation and proof of the geometric rule pre.
There are no restrictions on the base eld, and all multiplicities arising are 1. The littlewoodrichardson rule, and related combinatorics. The aim of this conference is to allow participants an opportunity to discuss recent developments and new ideas in algebraic and enumerative combinatorics and related areas such as representation theory, algebraic geometry and statistical physics. Math 99r, algebraic combinatorics symmetric functions. Skew littlewoodrichardson rules from hopf algebras.
The paper is due 2 days prior to the end of reading period, may 4 this is sharp the grades for the class are due may 6. The littlewoodrichardson rule, and related combinatorics core. Intheprequeltothispaper,weshowedhowresultsofmasoninvolving a new combinatorial formula for polynomials that are now known as demazure atoms characters of quotients of demazure modules, called stan. We use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine hecke algebra. In the past decade the littlewoodrichardson rule lr rule has moved into center stage in the combinatorics of young tableaux. Note that the content of a littlewoodrichardson tableau is always a partition. The main result of this paper is a combinatorial rule for the calculation of the littlewoodrichardson polynomials which provides a manifestly positive formula in the sense that c. Nov 16, 2011 the odd littlewoodrichardson rule article pdf available in journal of algebraic combinatorics 374 november 2011 with reads how we measure reads. Many popular objects in combinatorics, such as young tableaux and schur symmetric polynomials, originally came from representation theory of general linear and symmetric groups. The littlewoodrichardson rule states that the product of two schur polynomials can be written as a finite weighted sum of schur polynomials. Discrete mathematics 72 1988 385390 385 northholland on the littlewoodrichardson rule in terms of lattice path combinatorics toshihiro watanabe department of applied mathematics, faculty of engineering, gifu university, yanagido 11, gifu, 50111, japan received 22 august 1986 revised 9 april 1987 this work presents a simple proof of the littlewoodrichardson rule. We also establish skew littlewoodrichardson rules for schur p and qfunctions and noncommu. Hall polynomials, inverse kostka polynomials and puzzles. Algebraic combinatorics has long since reached the boundary of the universe of explicit exact formulas.
The rst is algebraic and geometric, connecting s nrepresentation theory and the cohomology ring of grassmannians. We conclude with a section relating these fairly modern techniques to earlier work on the littlewoodrichardson rule. The archetypal example of such a formula is the littlewoodrichardson rule, which gives the mul. Algebraic and enumerative combinatorics in okayama r esum e 5 a factorization formula for kkschur functions we give a pieritype formula for the sum of kkschur functions. Yet major problems from statistical mechanics to gct remain unsolved as they dont fall into the special. Pages in category algebraic combinatorics the following 46 pages are in this category, out of 46 total. In mathematics, the littlewoodrichardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two schur functions as a linear combination of other schur functions.
Schubert problems are among the most classical problems in enumerative geometry of. Call the number so associated to each entry its index, and write an entry i with index j as ij. The ams bookstore is open, but rapid changes related to the spread of covid19 may cause delays in delivery services for print products. The course will focus on combinatorics of root systems, weyl groups, and related structures. This volume consisting of two research papers and one survey paper is a good guide to look into a new emerging field, which stems from the interaction of combinatorics and representation theory. Proof of the geometric littlewoodrichardson rule theorem 2. A geometric littlewoodrichardson rule by ravi vakil abstract we describe a geometric littlewoodrichardson rule, interpreted as deforming the intersection of two schubert varieties into the union of schubert varieties. A littlewoodrichardson rule is a combinatorial interpretation of the littlewood. As in volume 1, the exercises play a vital role pdf in developing the material. In particular, we prove a version of the littlewoodrichardson rule for skew schur functions that was conjectured by assaf and mcnamara am, and which extends their skew pieri formula. This book is truly the only text i have ever read that has this perspective on the subject.
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