YOUNG TABLEAUX: WITH APPLICATIONS TO REPRESENTATION THEORY AND GEOMETRY (LMS Student Texts 35) By William Fulton: 260 pp., £14.95 (LMS Members' price £11.20), ISBN 0 521 56724 6 (Cambridge University Press, 1997)
YOUNG TABLEAUX: WITH APPLICATIONS TO REPRESENTATION THEORY AND GEOMETRY (LMS Student Texts 35) By William Fulton: 260 pp., £14.95 (LMS Members' price £11.20), ISBN 0 521 56724 6 (Cambridge University Press, 1997)
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DOI：
10.1112/s0024609398214779
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发表时间：
1998-09
影响因子：
0.9
通讯作者：
A. O. Morris
中科院分区：
文献类型：
--
作者：
A. O. Morris
组合学常常被认为是内省的并且缺乏与数学其他部分的联系。本书的主题“年轻的画面”当然是一个例外。这些纯粹的组合对象是由杰出而古怪的英国牧师阿尔弗雷德·杨（Alfred Young）在一系列才华横溢但几乎难以理解的论文中引入的，题为“论定量替代分析”，该论文发表在《Proc》杂志上。伦敦数学。苏克。 1899 年至 1934 年间，最后一本书是在他去世 12 年后的 1952 年追加的。这些论文本质上是完全组合的，尽管它们是以 19 世纪末流行的风格编写的经典不变量理论的自然发展。杨几乎没有想到他的作品会产生影响，不仅在数学的各个方向，包括组合数学，而且在很长一段时间内对物理学乃至其他科学分支产生影响。如果说杨的论文难以理解，那么他的许多继任者的著作也同样难以理解。此外，后来的、更可靠的解释者的纯粹主义方法也同样令人畏惧。本书的作者不仅因其在数学领域广泛的重大基础贡献而闻名，而且还是一位出色的解释者。这个主题一定是一个重大挑战。就其本质而言，组合数学中的问题通常通过查看足够普遍的例子很容易理解，但一旦形式化，它们就会变得深不可测。本书优于许多其他组合书籍的地方在于，它成功地在示例和一般演示之间实现了巨大的飞跃。年轻的画面很容易描述：它们只是排列在左对齐行中的 n 个盒子，里面填满了数字 1、2、...、n。这些简单的物体被证明具有如此巨大的价值并且在数学中如此普遍，这几乎是一个奇迹。多年来，人们开发了许多新的、原创的方法，为该主题提供了新的见解和方向——下面列出了一些更重要的早期贡献者。本书的第 1 部分描述了其中两种方法：Schensted“碰撞”算法（它导致了 Robinson-Schensted-Knuth 算法）和 Schutzenberger（jeu de taquin）“滑动”算法。这些导致了同样基本且广泛应用的利特尔伍德-理查森“乘法”杨氏场景规则的两个证明。尽管自该规则首次提出以来已经过去了 60 多年，但第一个证据只是在最近才出现。这里提出了一个新的证明，它被正确地称为迄今为止最简单的证明。给出了对称多项式，特别是 Schur 多项式的简短独立介绍。接下来是对称群表示论的描述，强调 Specht 模方法和连接
Combinatorics is often claimed to be introspective and to lack contact with other parts of mathematics. The subject of this book, Young tableaux, is certainly an exception. These purely combinatorial objects were introduced by the remarkable and eccentric English clergyman Alfred Young, in a brilliant but almost impenetrable series of papers entitled ‘On quantative substitutional analysis ’ which appeared in Proc. London Math. Soc. between 1899 and 1934, with a final one added posthumously in 1952, twelve years after his death. These papers were totally combinatorial in nature, although a natural development of the classical theory of invariants written in the style which was in vogue towards the end of the 19th century. Little would Young have guessed the impact his tableaux would make, not only in various directions in mathematics, including combinatorics, but also over a long period in physics and, indeed, in other branches of science. If Young’s papers are difficult to understand, the same is true about many of his successors’ works, which could be equally inaccessible. Furthermore, the purist approach of later, more reliable, expositors can be just as daunting. The author of this book is well known not only for his major fundamental contributions broadly within mathematics, but also as a superb expositor. This subject must have been a major challenge. By their nature, problems in combinatorics are often easily understood through looking at sufficiently general examples—but once formalised they can become unfathomable. Where this book scores over many other combinatorial books is that it has successfully made the quantum leap between example and a general presentation. Young tableaux are easily described: they are simply n boxes laid out in leftjustified rows, filled with the numbers 1, 2,..., n. That such simple objects have proved to be of so great value and are so pervasive in mathematics is almost miraculous. Over the years, many new and original approaches to them have been developed, giving new insights and directions to the subject—some of the more significant earlier contributors are named below. Two of these approaches are described in Part 1 of the book: the Schensted ‘bumping’ algorithm, which leads to the Robinson–Schensted– Knuth algorithm, and the Schutzenberger ( jeu de taquin) ‘ sliding’ algorithm. These lead to two proofs of the equally fundamental and widely applied Littlewood– Richardson rule for ‘multiplying’ Young tableaux. Although over 60 years have elapsed since this rule was first stated, the first proofs appeared only relatively recently. Here a new proof is presented, which is rightly claimed to be the easiest yet. A brief self-contained introduction to symmetric polynomials, and especially Schur polynomials, is given. This is followed by a description of the representation theory of symmetric groups, emphasising the Specht module approach and the connection