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1	Symmetric functions and Macdonald polynomials Langer, R. January 2008 (has links) The ring of symmetric functions Λ, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on Λ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood–Richardson numbers may be viewed as the structure constants for the co-product in the Schur basis. The first part of this thesis, inspired by the umbral calculus of Gian-Carlo Rota, is a study of the co-algebra maps of Λ. The Macdonald polynomials are a somewhat mysterious qt-deformation of the Schur functions. The second part of this thesis contains a proof a generating function identity for the Macdonald polynomials which was originally conjectured by Kawanaka.
2	A Puzzle-Based Synthesis Algorithm For a Triple Intersection of Schubert Varieties Brown, Andrew Alexander Harold 28 January 2010 (has links) This thesis develops an algorithm for the Schubert calculus of the Grassmanian. Specifically, we state a puzzle-based, synthesis algorithm for a triple intersection of Schubert varieties. Our algorithm is a reformulation of the synthesis algorithm by Bercovici, Collins, Dykema, Li, and Timotin. We replace their combinatorial approach, based on specialized Lebesgue measures, with an approach based on the puzzles of Knutson, Tao and Woodward. The use of puzzles in our algorithm is beneficial for several reasons, foremost among them being the larger body of work exploiting puzzles. To understand the algorithm, we study the necessary Schubert calculus of the Grassmanian to define synthesis. We also discuss the puzzle-based Littlewood-Richardson rule, which connects puzzles to triple intersections of Schubert varieties. We also survey three combinatorial objects related to puzzles in which we include a puzzle-based construction, by King, Tollu, and Toumazet, of the well known Horn inequalities. Schubet Calculus Littlewood-Richardson rule Puzzle Synthesis Combinatorics and Optimization
3	A Puzzle-Based Synthesis Algorithm For a Triple Intersection of Schubert Varieties Brown, Andrew Alexander Harold 28 January 2010 (has links) This thesis develops an algorithm for the Schubert calculus of the Grassmanian. Specifically, we state a puzzle-based, synthesis algorithm for a triple intersection of Schubert varieties. Our algorithm is a reformulation of the synthesis algorithm by Bercovici, Collins, Dykema, Li, and Timotin. We replace their combinatorial approach, based on specialized Lebesgue measures, with an approach based on the puzzles of Knutson, Tao and Woodward. The use of puzzles in our algorithm is beneficial for several reasons, foremost among them being the larger body of work exploiting puzzles. To understand the algorithm, we study the necessary Schubert calculus of the Grassmanian to define synthesis. We also discuss the puzzle-based Littlewood-Richardson rule, which connects puzzles to triple intersections of Schubert varieties. We also survey three combinatorial objects related to puzzles in which we include a puzzle-based construction, by King, Tollu, and Toumazet, of the well known Horn inequalities. Schubet Calculus Littlewood-Richardson rule Puzzle Synthesis Combinatorics and Optimization
4	Réduction des graphes de Goresky-Kottwitz-MacPherson ; nombres de Kostka et coefficients de Littlewood-Richardson Cochet, Charles 19 December 2003 (has links) (PDF) Ce travail concerne la réalisation concrète en calcul formel d'algorithmes abstraits issus de publications récentes. Il comporte deux parties distinctes mais cependant issues du m(ê)me monde : l'action d'un groupe de Lie, sur une variété ou un espace vectoriel. La première partie traite de l'implémentation de la réduction d'un graphe de Goresky-Kottwitz-MacPherson. Ce graphe est l'analogue combinatoire d'une variété symplectique compacte connexe soumise à une action hamiltonienne d'un tore compact. La seconde partie est consacrée à l'implémentation du calcul de deux coefficients intervenant lors de l'action d'un groupe de Lie semi-simple complexe sur un espace vectoriel de dimension finie : la multiplicité d'un poids dans une représentation irréductible de dimension finie (nombre de Kostka) et les coefficients de décomposition du produit tensoriel de deux représentations irréductibles de dimension finie (coefficients de Littlewood-Richardson). [MATH] Mathematics graphe de Goresky-Kottwitz-MacPherson réduction symplectique nombre de Kostka coefficient de Littlewood-Richardson fonction de partition vectorielle polytope
5	Diameter of a Rouquier block Mayer, Andrew 14 June 2018 (has links) No description available. Mathematics Theoretical Mathematics Rouquier block blocks diameter graph degree Littlewood-Richardson Littlewood Richardson weight abacus Young diagram representation partition
6	A LOWER BOUND ON THE DISTANCE BETWEEN TWO PARTITIONS IN A ROUQUIER BLOCK Bellissimo, Michael Robert 08 June 2018 (has links) No description available. Mathematics partitions badness Rouquier Block representation theory combinatorics Brauer Graph diameter of a Brauer Graph partition theory Young Diagram decomposition matrix p-core abacus p-regular partitions Littlewood-Richardson Rule
7	On the Diameter of the Brauer Graph of a Rouquier Block of the Symmetric Group Trinh, Megan 08 June 2018 (has links) No description available. Mathematics mathematics partitions combinatorics Rouquier block abacus Young diagram weight quotient Littlewood-Richardson Rule Chuang-Tan Rule Brauer graphs hook length rim hook representation theory

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