伊赛·舒尔Issai Schur

伊赛·舒尔出生在第聂伯河畔的莫吉廖夫，他说德语时没有一丝口音，甚至没有人猜到德语不是他的第一语言。他13岁时去了拉脱维亚，在那里就读于Libau的文理中学，该地现在称为Liepaja。

1894年，舒尔进入柏林大学学习数学和物理。费迪南德·格奥尔格·弗罗贝尼乌斯是他的老师之一，并且对舒尔产生了很大影响，后来还指导了他的博士研究。费迪南德·格奥尔格·弗罗贝尼乌斯和威廉·伯恩赛德是theory of representations of groups的两个主要创始人，作为matrices的群。这一理论被证明是研究群的一个非常强大的工具，而舒尔将从费迪南德·格奥尔格·弗罗贝尼乌斯那里学习这门学科的基础。舒尔随后取得了重大进展，既包括他自己的工作，也包括与费迪南德·格奥尔格·弗罗贝尼乌斯合作完成的工作。

1901年，舒尔凭借一篇研究复数域上一般线性群的有理表示的学位论文获得博士学位。舒尔在其学位论文中引入的函数今天被称为$S$函数，其中$S$代表舒尔。对舒尔学位论文结果的兴趣延续至今；例如，詹姆斯·亚历山大·格林于1980年发表了这些结果在现代框架下的论述。

1903年，舒尔成为柏林大学讲师，随后从1911年到1916年在波恩大学担任数学教授。1916年他回到柏林，在那里建立了著名的学派，并在那里度过了余生的大部分时间。1919年，即回到柏林三年后，他被提升为柏林的正教授，并一直担任此讲席，直到1935年被纳粹解职。

舒尔主要以他在群表示论方面的基础性工作闻名，但他也在数论、分析以及下文描述的其他主题上工作过。1904年至1907年间，他研究群的射影表示和group characters。他此时发现的最基础性的结果之一今天称为舒尔引理。

在一系列论文中，他引入了现在称为“舒尔乘子”的概念。这是一个极其重要的抽象概念，源于舒尔当时研究的具体问题。很久以后，1949年，塞缪尔·艾伦伯格和桑德斯·麦克兰恩定义了cohomology groups。他们当时不知道，系数在非零复数中的第二上同调群就是舒尔乘子，因此舒尔在四十年前已经迈出了最初的一些步骤。

大约1914年，舒尔对群表示的兴趣被搁置一旁，他转而研究其他主题，但大约1925年，理论物理学的发展表明群表示在该学科中具有根本重要性。舒尔以 renewed vigour 重新投入表示论工作，得以完成始于其博士论文的研究计划，并给出一般线性群有理表示的完整描述。

舒尔还对可约性、根的定位以及多项式类（如埃德蒙·拉盖尔和夏尔·埃尔米特多项式）的Galois group的构造感兴趣。在[1]中，给出了舒尔所研究的其他主题的提示：-

首先是纯群论，其中舒尔采用了令人惊讶的方法，即在不借助特征标的情况下证明此前只能通过特征标来证明的定理。
其次，他研究矩阵领域。
第三，他处理代数方程，有时进行根的求值，有时处理所谓的无影响方程，即具有对称埃瓦里斯特·伽罗瓦群的方程。他也是第一个给出具有alternating伽罗瓦群的方程例子的人。
第四，他研究数论；
第五，研究发散级数；
第六，研究积分方程；
最后是函数论。

舒尔在柏林建立的学派不仅对群的表示论具有重大意义，而且如上所述，对数学的其他领域也是如此。该学派部分通过舒尔的讲座[7]开展工作：-

……有[许多]数学家在柏林听过舒尔的讲座和讨论班，并深受他的影响……

该学派还开展了合作[1]：-

与许多同事的活跃交流促使舒尔撰写了重要的回忆录……其中一些是与其他作者合作发表的，尽管在当时双署名的出版物几乎闻所未闻。

这个学派无疑是柏林最具连贯性和影响力的数学家群体，也是全德国最重要的群体之一。舒尔富有魅力的领导激励着周围的人推进群表示论的研究。舒尔本人令人瞩目的贡献被他的学生们向若干不同方向加以拓展。他们研究的课题包括soluble groups、组合学和矩阵论。

在舒尔指导下完成博士学位的学生中有理查德·布饶尔、阿尔弗雷德·布劳尔（理查德·布饶尔的兄弟）、Robert Frucht、伯恩哈德·诺伊曼、理查德·拉多和Helmut Wielandt。还有其他人曾在舒尔指导下工作，如Kurt Hirsch、Walter Ledermann、Hanna Neumann和Menahem Max Schiffer。

[7]中的Ledermann将舒尔描述为一位教师：-

舒尔是一位极出色的讲师。他的讲座经过精心准备……[并且]极受欢迎。我记得曾去听他的代数课，那是在一个坐满了大约400名学生的讲堂里进行的。有时，当我只能满足于坐在讲堂后排的座位时，我就用一副观剧望远镜，至少能瞥见演讲者一眼。

1922年，舒尔经科学院秘书马克斯·普朗克提议，当选为柏林科学院院士。马克斯·普朗克那篇列举舒尔杰出成就的致辞，是由费迪南德·格奥尔格·弗罗贝尼乌斯撰写的，至少写于五年前，因为费迪南德·格奥尔格·弗罗贝尼乌斯已于1917年去世。

从1933年起，德国发生的事件使舒尔的生活日益艰难。Hirsch谈到1933年4月1日的事件，当时标语上写着“德国人保卫自己，反对犹太暴行宣传：只在德国商店购物”：-

那就是所谓的“抵制日”，那一天犹太商店遭到抵制，犹太教授和讲师不被允许进入大学。在场的每个人都必须就德国的复兴等话题发表简短讲话。路德维希·比贝尔巴赫讲得相当得体，然后他说：“一滴悔恨落入我的喜悦之中，因为我亲爱的朋友和同事舒尔今天不被允许与我们在一起。”

1933年4月7日，纳粹通过了一项法律，其第三条规定，非雅利安血统的公务员必须退休，但参加过第一次世界大战的人和战前官员可获豁免。舒尔在第一次世界大战前曾担任职务，本应使他具备公务员资格，但事实不被允许妨碍这一决定，他还是被“退休”了。Schiffer写道[8]：-

舒尔的课程被取消，学生和教授中爆发了抗议，因为舒尔受人尊敬，非常受人喜爱。第二天，艾尔哈德·施密特以抗议这次解职开始了他的讲座，甚至路德维希·比贝尔巴赫——他后来作为纳粹分子落得可耻名声——也站出来为舒尔辩护。舒尔则在家中平静地继续他的代数工作。

舒尔视自己为德国人，而非犹太人，无法理解他在纳粹统治下遭受的迫害和羞辱。事实上，舒尔的解职被撤销，他得以在一段时间内履行部分职责。到1933年11月，Walter Ledermann参加他的国家考试（Staatsexamen）时，他由舒尔和身穿纳粹制服的路德维希·比贝尔巴赫一起考核。

有人邀请舒尔去美国和英国，但他全都拒绝了，无法理解一个德国人怎么会在德国不受欢迎。例如，Ledermann在1934年春天获得了一项去苏格兰圣安德鲁斯的奖学金，他试图说服舒尔和他一起去圣安德鲁斯，但没有成功。

舒尔继续遭受强加于他的屈辱。Schiffer回忆了[8]中一件与1935年1月10日舒尔六十岁生日有关的事：-

舒尔告诉我，柏林数学研究所里唯一对他好的人是Grunsky，当时是一位年轻讲师。战后很久，我和Grunsky谈起这句话，他真的哭了起来：“你知道我做了什么吗？我给他寄了一张明信片祝贺他六十岁生日。我非常钦佩他，在那张卡片上非常恭敬。他一定非常孤独，才会记得这么一件小事。”

1935年晚些时候，舒尔被解除柏林讲席，但他继续在那里工作，忍受着巨大的艰辛和困难。阿尔弗雷德·布劳尔在[6]中写道：-

当埃德蒙·朗道于1938年2月去世时，舒尔本应在葬礼上致辞。为此，他需要从文献中获取一些数学细节。他请我在这件事上帮助他。当然，我不被允许使用我多年来建立的数学研究所图书馆。最后我获得了一周的豁免，可以付费使用普鲁士国家图书馆。……因此我至少能回答舒尔的一些问题。

有人向舒尔施压，要求他从柏林科学院辞职，他曾在1922年荣幸地当选为该会成员。1938年3月29日，路德维希·比贝尔巴赫在柏林科学院的一份文件上舒尔的签名下方写道：-

我觉得奇怪，犹太人仍然是学术委员会的成员。

仅仅一个多星期后，1938年4月7日，舒尔辞去了科学院的多个委员会职务。然而，对他的压力仍在继续，同年晚些时候，他完全辞去了科学院的职务。

舒尔于1939年离开德国前往巴勒斯坦，身心俱疲，还遭受了最后的屈辱：被迫寻找担保人支付“帝国逃亡税”，才获准离开德国。由于没有足够的资金在巴勒斯坦生活，他被迫将自己心爱的学术书籍卖给普林斯顿的爱德华·斯图迪高等研究院。两年后，他在66岁生日那天去世。

Although Issai Schur was born in Mogilev on the Dnieper, he spoke German without a trace of an accent, and nobody even guessed that it was not his first language. He went to Latvia at the age of 13 and there he attended the Gymnasium in Libau, now called Liepaja.

In 1894 Schur entered the University of Berlin to read mathematics and physics. Frobenius was one of his teachers and he was to greatly influence Schur and later to direct his doctoral studies. Frobenius and Burnside had been the two main founders of the theory of representations of groups as groups of matrices. This theory proved a very powerful tool in the study of groups and Schur was to learn the foundations of this subject from Frobenius. Schur then made major steps forward, both in work of his own and work done in collaboration with Frobenius.

In 1901 Schur obtained his doctorate with a thesis which examined rational representations of the general linear group over the complex field. Functions which Schur introduced in his thesis are today called $S$-functions, where the $S$ stands for Schur. Interest in the results of Schur's thesis continues today; for example J A Green published an account of these results in a modern setting in 1980.

In 1903 Schur became a lecturer at Berlin University and then, from 1911 until 1916, he held a professorship in mathematics at the University of Bonn. He returned to Berlin in 1916 and there he built his famous school and spent most of the rest of his life there. He was promoted to full professor in Berlin in 1919, three years after he returned there, and he held this chair until he was dismissed by the Nazis in 1935.

Schur is mainly known for his fundamental work on the representation theory of groups but he also worked in number theory, analysis and other topics described below. Between 1904 and 1907 he worked on projective representations of groups and group characters. One of the most fundamental results which he discovered at this time is today called Schur's Lemma.

In a series of papers he introduced the concept now known as the 'Schur multiplier'. This is an extremely important abstract concept which arose from the concrete problems that Schur was studying. Much later, in 1949, Eilenberg and Mac Lane defined cohomology groups. They were unaware at that time that the second cohomology group with coefficients in the nonzero complex numbers is the Schur multiplier, and therefore that Schur had made some of the first steps forty years earlier.

Around 1914 Schur's interest in representations of groups was put to one side while he worked on other topics but, around 1925, developments in theoretical physics showed that group representations were of fundamental importance in that subject. Schur returned to work on representation theory with renewed vigour and he was able to complete the programme of research begun in his doctoral dissertation and give a complete description of the rational representations of the general linear group.

Schur was also interested in reducibility, location of roots and the construction of the Galois group of classes of polynomials such as Laguerre and Hermite polynomials. In [1] an indication of the other topics that Schur worked on is given:-

First there was pure group theory, in which Schur adopted the surprising approach of proving without the aid of characters, theorems that had previously been demonstrated only by that means.
Second, he worked in the field of matrices.
Third, he handled algebraic equations, sometimes proceeding to the evaluation of roots, and sometimes treating the so-called equation without affect, that is, with symmetric Galois groups. He was also the first to give examples of equations with alternating Galois groups.
Fourth, he worked in number theory;
Fifth, in divergent series;
Sixth, in integral equations;
and lastly in function theory.

The school which Schur built at Berlin was of major importance not only for the representation theory of groups but, as indicated above, for other areas of mathematics. The school partly worked through the Schur's lecturing [7]:-

...there are [many] mathematicians who went to Schur's lectures and seminars in Berlin and were strongly influenced by him...

The school also worked with collaborations [1]:-

A lively interchange with many colleagues led Schur to contribute important memoirs .... Some of these were published as collaborations with other authors, although publications with dual authorship were almost unheard of at that time.

This school was certainly the most coherent and influential group of mathematicians in Berlin, and among the most important in all of Germany. Schur's charismatic leadership inspired those around him to push forward with research on group representations. Schur's own impressive contributions were extended by his students in a number of different directions. They worked on topics such as soluble groups, combinatorics, and matrix theory.

Among the students who completed their doctorates under Schur were Richard Brauer, Alfred Brauer (Richard Brauer's brother), Robert Frucht, Bernhard Neumann, Richard Rado, and Helmut Wielandt. There were others who worked under Schur such as Kurt Hirsch, Walter Ledermann, Hanna Neumann and Menahem Max Schiffer.

Ledermann in [7] describes Schur as a teacher:-

Schur was a superb lecturer. His lectures were meticulously prepared... [and] were exceedingly popular. I remember attending his algebra course which was held in a lecture theatre filled with about 400 students. Sometimes, when I had to be content with a seat at the back of the lecture theatre, I used a pair of opera glasses to get at least a glimpse of the speaker.

In 1922 Schur was elected to the Berlin Academy, proposed by Planck, the secretary of the Academy. Planck's address which listed Schur's outstanding achievements had been written by Frobenius, at least five years earlier, as Frobenius died in 1917.

From 1933 events in Germany made Schur's life increasingly difficult. Hirsch spoke of the events of 1 April 1933 when posters carried the message 'Germans defend yourselves against jewish atrocity propaganda : buy only at German shops':-

That was the so-called 'Boycott Day', the day on which Jewish shops were boycotted and Jewish professors and lecturers were not allowed to enter the university. Everybody who was there had to make a little speech about the rejuvenation of Germany etc. And Bieberbach did this quite nicely and then he said 'A drop of remorse falls into my joy because my dear friend and colleague Schur is not allowed to be among us today'.

On 7 April 1933 the Nazis passed a law which, under clause three, ordered the retirement of civil servants who were not of Aryan descent, with exemptions for participants in World War I and pre-war officials. Schur had held an appointment before World War I which should have qualified him as a civil servant, but the facts were not allowed to get in the way, and he was 'retired'. Schiffer wrote [8]:-

When Schur's lectures were cancelled there was an outcry among the students and professors, for Schur was respected and very well liked. The next day Erhard Schmidt started his lecture with a protest against this dismissal and even Bieberbach, who later made himself a shameful reputation as a Nazi, came out in Schur's defence. Schur went on quietly with his work on algebra at home.

Schur saw himself as a German, not a Jew, and could not comprehend the persecution and humiliation he suffered under the Nazis. In fact Schur's dismissal was revoked and he was able to carry out some of his duties for a while. By November 1933 when Walter Ledermann took his Staatsexamen he was examined by Schur together with Bieberbach who was wearing Nazi uniform.

There were invitations to Schur to go to the United States and to Britain but he declined them all, unable to understand how a German was not welcome in Germany. For example Ledermann obtained a scholarship to go to St Andrews in Scotland in the spring of 1934 and he tried unsuccessfully to persuade Schur to join him in St Andrews.

Schur continued to suffer the humiliation that was heaped on him. Schiffer recalls an event in [8] relating to Schur's 60th birthday on 10 January 1935:-

Schur told me that the only person at the Mathematical Institute in Berlin who was kind to him was Grunsky, then a young lecturer. Long after the war, I talked to Grunsky about that remark and he literally started to cry: "You know what I did? I sent him a postcard to congratulate him on his sixtieth birthday. I admired him so much and was very respectful in that card. How lonely he must have been to remember such a small thing."

Later in 1935 Schur was dismissed from his chair in Berlin but he continued to work there suffering great hardship and difficulties. Alfred Brauer writes in [6]:-

When Landau died in February 1938, Schur was supposed to give an address at his funeral. For that reason he was in need of some mathematical details from the literature. He asked me to help him in this matter. Of course I was not allowed to use the library of the mathematical institute which I had built up over many years. Finally I got an exemption for a week and could use the library of the Prussian Staatsbibliothek for a fee. ... So I could answer at least some of Schur's questions.

Pressure was put on Schur to resign from the Berlin Academy to which he had been honoured to be elected in 1922. On 29 March 1938 Bieberbach wrote below Schur's signature on a document of the Berlin Academy:-

I find it surprising that Jews are still members of academic commissions.

Just over a week later, on 7 April 1938, Schur resigned from Commissions of the Academy. However, the pressure on him continued and later that year he resigned completely from the Academy.

Schur left Germany for Palestine in 1939, broken in mind and body, having the final humiliation of being forced to find a sponsor to pay the 'Reichs flight tax' to allow him to leave Germany. Without sufficient funds to live in Palestine he was forced to sell his beloved academic books to the Institute for Advanced Study in Princeton. He died two years later on his 66th birthday.