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タイトル: Pieri's Formula for Quantum Schubert Polynomials Corresponding to Grassmannian Permutations
著者: Watanabe, Yurika
著者(別表記) : 渡邊, 百合佳
著者読み: わたなべ, ゆりか
キーワード: Pieri’s Formula; Quantum Schubert Polynomial; quantum Schur functions.; Grassmannian Permutation
発行日: 2015年 3月31日
出版者: 奈良女子大学大学院人間文化研究科
引用: 渡邊百合佳:人間文化研究科年報(奈良女子大学大学院人間文化研究科), 第30号, pp.113-123
抄録: The subject of this study is “Pieri's Formula via the Young Diagrams” for the quantum Schur functions. The flag manifold can be decomposed into even-dimensional cells― Schubert cells―and indexed by the elements of the symmetric group. By the Poincaré duality, we associate the closure of the Schubert cells or Schubert varieties, with the Schubert classes in the cohomology of the flag manifold. The set of Schubert classes is an additive basis of the integral cohomology ring of the flag manifold. Although we know the simplest multiplication formula, the “Monk formula,” there is, as yet, no general formula for multiplication. On the other hand, the cohomology ring of the flag manifold is isomorphic to the coinvariant ring of the symmetric group; there exists polynomial representatives for the Schubert classes with remarkable properties, which is called the “Schubert polynomials.” Furthermore, we can quantize the Schubert polynomials to describe the small quantum cohomology of the flag manifolds. A quantum version of Pieri's formula for the flag manifolds is known in terms of quadratic algebra, and we have an explicit formula for multiplying Schubert classes in the small quantum cohomology of the Grassmannians. Since the quantum cohomology is not functorial, the formulas for the Grassmannians are not the same as those for the flag manifolds. Therefore, we study the multiplication formulas for the quantum Schubert polynomials corresponding to Grassmannian permutations which are the quantum Schur functions, and we obtain explicit formulas for the quantum Schur functions corresponding to hooks of arm length one and Young diagrams with one row or one column.
URI: http://hdl.handle.net/10935/3971
ISSN: 09132201
出現コレクション:第30号

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