费迪南德·格奥尔格·弗罗贝尼乌斯Ferdinand Georg Frobenius

弗罗贝尼乌斯的父亲是Christian 费迪南德·格奥尔格·弗罗贝尼乌斯，一位新教牧师，母亲是Christine Elizabeth Friedrich。弗罗贝尼乌斯出生在夏洛滕堡，这是柏林的一个区，直到1920年才并入该市。他于1860年进入费迪南德·约阿希姆斯塔尔文理中学，当时他将近十一岁，并于1867年毕业。同年，他前往哥廷根大学开始大学学习，但只在那里学习了一个学期就回到了柏林。

回到柏林大学后，他听了利奥波德·克罗内克、恩斯特·爱德华·库默尔和卡尔·魏尔斯特拉斯的讲座。他继续在那里攻读博士学位，参加恩斯特·爱德华·库默尔和卡尔·魏尔斯特拉斯的讨论班，并于1870年在卡尔·魏尔斯特拉斯的指导下获得博士学位（以优异成绩授予）。1874年，在先后在约阿希姆斯塔尔 文理中学和Sophienrealschule中学任教后，他被任命为柏林大学的数学编外教授。

关于弗罗贝尼乌斯迄今为止的职业生涯的描述，细心的读者可能已经注意到，没有提到他在被任命为教职之前获得了教授资格论文（Habilitation）。这不是遗漏，而是考虑到德国制度的严格性，这种情况被允许是令人惊讶的。这次任命的细节在[3]中给出，但我们应该说，这最终之所以成为可能，必定是由于卡尔·魏尔斯特拉斯的大力支持，他极具影响力，并认为弗罗贝尼乌斯是他最有天赋的学生之一。

弗罗贝尼乌斯只在柏林待了一年，就前往苏黎世，接受了瑞士联邦理工学院常任教授的任命。从1875年到1892年的十七年间，弗罗贝尼乌斯在苏黎世工作。他在那里结婚成家，并在数学的广泛不同领域中做了许多重要工作。我们将在下面讨论他研究过的一些主题，但目前我们将继续描述弗罗贝尼乌斯的职业生涯是如何发展的。

1891年12月的最后几天，利奥波德·克罗内克去世，因此他在柏林的讲席空缺。卡尔·魏尔斯特拉斯坚信弗罗贝尼乌斯是使柏林保持在数学前沿的合适人选，利用他的巨大影响力促成了弗罗贝尼乌斯的任命。然而，由于我们稍后将讨论的原因，弗罗贝尼乌斯对柏林大学的数学来说结果喜忧参半。

他被任命的积极一面无疑是他在表示论的群方面做出的卓越贡献，特别是他对特征标理论的发展，以及他作为当时顶尖数学家之一的地位。消极的一面主要源于他的个性，[5]中将其描述为：-

……偶尔易怒、好争吵，并且喜欢谩骂。

Biermann在[3]中更仔细地审视了他的性格（没有双关之意！），以及它如何影响了大学数学教育的成效。他描述了弗罗贝尼乌斯与他在柏林同事之间发展出的紧张关系。他的标准如此之高，最终这些标准并未给柏林带来好处。他[3]：-

……一有机会就怀疑部里有意降低柏林大学的标准，用弗罗贝尼乌斯的话说，降到技术学校的水平……即便如此，拉扎勒斯·福克斯和赫尔曼·阿曼杜斯·施瓦茨还是顺从他，后来弗里德里希·赫曼·肖特基也是如此，后者仅因他的召唤才来到柏林。弗罗贝尼乌斯是核心人物，柏林大学数学的命运在他身上维系了25年。当然，他并非没有注意到，博士学位、教授资格论文（Habilitation）和讲师（Docent）的数量缓慢但确定地下降，尽管学生人数大幅增加。他无法阻止这一点，无法实现保持卡尔·魏尔斯特拉斯、恩斯特·爱德华·库默尔和利奥波德·克罗内克时代的外在面貌不变的目标，只能无助地目睹这些发展，对于他易怒的性格来说，这是双重难以忍受的。

我们不应过于苛责弗罗贝尼乌斯，因为正如Haubrich在[5]中所解释的，弗罗贝尼乌斯的态度是当时柏林所有数学教授典型的态度：-

他们都深感有义务延续普鲁士新人文主义的大学研究与教学传统，正如他们自己作为学生时所经历的那样。这对弗罗贝尼乌斯尤其如此。他认为自己是一位学者，其职责是为纯数学的知识做出贡献。在他看来，应用数学属于技术学院。

然而，哥廷根大学的数学观却截然不同。当时柏林大学和哥廷根大学的数学家之间存在竞争，但这场竞争哥廷根赢了，因为在菲利克斯·克莱因的领导下，那里的数学蓬勃发展，这让弗罗贝尼乌斯非常恼火。在[3]中，Biermann写道：-

弗罗贝尼乌斯对菲利克斯·克莱因和索菲斯·李的厌恶没有止境……

弗罗贝尼乌斯憎恨哥廷根所代表的数学风格。这是一种新方法，标志着与德国大学传统风格的显著变化。如上所述，弗罗贝尼乌斯持有极其传统的观点。在1896年2月3日写给阿道夫·赫维兹（哥廷根体系的产物）的一封信中（见[4]）：-

如果你是从一个更爱玩弄玫瑰色意象而非硬思想的学派中出来的，并且如果令我高兴的是，你也在逐渐从中解放出来，那么旧爱不会生锈。请把这玩笑当作玩笑。

然而，应该看到另一面，因为在[9]中，卡尔·西格尔从1915年成为学生起认识弗罗贝尼乌斯两年，直到弗罗贝尼乌斯去世，他描述了对弗罗贝尼乌斯的印象，认为他个性热情，并表达了对他的快节奏、多样且深刻讲座的欣赏。其他人则会将他的讲座描述为扎实但不刺激。

为了了解弗罗贝尼乌斯在1892年被任命到柏林之前的工作质量，我们最好查看卡尔·魏尔斯特拉斯和拉扎勒斯·福克斯在1892年弗罗贝尼乌斯当选为柏林科学院成员时的推荐信。在[4]中给出了该文件以及来自拉扎勒斯·福克斯和赫尔曼·冯·亥姆霍兹的另一份类似文件的相当广泛的引文，但我们引用一小段摘录，以展示弗罗贝尼乌斯在苏黎世时期工作的力量、多样性和高质量。卡尔·魏尔斯特拉斯和拉扎勒斯·福克斯列出了弗罗贝尼乌斯做出重大贡献的15个主题：-

1. On the development of analytic functions in series。
2. On the algebraic solution of equations, whose coefficients are 有理函数 of one variable。
3. The theory of linear 微分方程。
4. On Pfaff's problem。
5. Linear forms with integer coefficients。
6. On linear substitutions and bilinear forms...
7. On adjoint linear differential operators...
8. The theory of elliptic and Jacobi functions...
9. On the relations among the 28 double tangents to a plane of degree 4。
10. On Sylow's theorem。
11. On double cosets arising from two finite groups。
12. On Jacobi's covariants...
13. On Jacobi functions in three variables。
14. The theory of biquadratic forms。
15. On the theory of surfaces with a differential parameter。

在群论方面的工作中，弗罗贝尼乌斯综合了代数方程理论、几何学和数论的结果，这引导他研究抽象群。他于1879年（与苏黎世的同事Stickelberger合作）发表了Über Gruppen von vertauschbaren ElementenⓉ（《论可交换元素的群》），该文考察了群中的可置换元素。这篇论文还给出了有限生成阿贝尔群的结构定理的证明。1884年，他发表了下一篇关于有限群的论文，其中他为抽象群证明了彼得·卢德维格·梅德尔·西罗的定理（彼得·卢德维格·梅德尔·西罗在其原始论文中是将该定理作为关于置换群的结果来证明的）。弗罗贝尼乌斯给出的证明基于共轭类，至今仍在大多数本科课程中使用。

在1887年的下一篇论文中，弗罗贝尼乌斯继续了他对群中共轭类的研究，这在他后来关于特征标的工作中将被证明是重要的。在这篇论文的引言中，他解释了自己是如何对抽象群产生兴趣的，这是通过研究利奥波德·克罗内克的一篇论文。然而，正是在1896年，当弗罗贝尼乌斯担任柏林大学教授时，他真正重要的群论工作才开始出现。那一年他发表了五篇关于群论的论文，其中一篇Über die Gruppencharactere关于群特征标的论文具有根本的重要性。他在这篇论文中写道：

我将在此发展[任意有限群的特征标]这一概念，相信通过引入它，群论将得到实质性的丰富。

这篇关于群特征标的论文于1896年7月16日提交给柏林科学院，其中包含弗罗贝尼乌斯在前几个月所进行的工作。在一系列写给理查德·戴德金的信中，第一封写于1896年4月12日，他关于群特征标的思想迅速发展。理查德·戴德金1885年一篇论文中的思想做出了重要贡献，弗罗贝尼乌斯得以构造出用复数表示的完整表示集。然而值得注意的是，尽管我们今天将弗罗贝尼乌斯关于群特征标的论文视为群表示论的根本性工作，但弗罗贝尼乌斯实际上在这项工作中引入群特征标时并未提及表示。直到第二年，群的表示才开始进入视野，这同样是弗罗贝尼乌斯提出的概念。因此，1897年是群表示论诞生之年。

在1897-1899年间，弗罗贝尼乌斯发表了两篇关于群表示的论文，一篇关于诱导特征标，一篇关于特征标的张量积。1898年，他引入了诱导表示的概念和弗罗贝尼乌斯互反定理。这是一阵活动爆发，为整个表示论机器奠定了基础。

在1896年4月26日给理查德·戴德金的一封信中，弗罗贝尼乌斯给出了交错群 $A_{4}, A_{5}$、对称群 $S_{4}, S_{5}$和168阶群$PSL(2,7)$的不可约特征标。他于1900年完全确定了对称群的特征标，并于1901年确定了交错群的特征标，分别发表了权威性论文。他在1900年和1901年的论文中继续应用特征标理论，研究了弗罗贝尼乌斯群的结构。

直到1897年，弗罗贝尼乌斯才得知特奥多尔·莫林的工作，他在给理查德·戴德金的一封信中将其描述为“非常美丽但困难”。他用matrices的语言重新表述了特奥多尔·莫林的工作，然后证明他的特征标是不可约表示的迹。这项工作于1897年发表。弗罗贝尼乌斯的特征标理论被威廉·伯恩赛德卓有成效地使用，并在威廉·伯恩赛德1911年版的Theory of Groups of Finite Order中得到了优美的阐述。

弗罗贝尼乌斯有许多对数学做出重要贡献的博士生。其中包括1899年获得博士学位的埃德蒙·朗道、1901年获得博士学位的伊赛·舒尔和1910年获得博士学位的罗伯特·雷马克。弗罗贝尼乌斯在群表示论和群特征标理论方面与伊赛·舒尔合作。弗罗贝尼乌斯如此迅速地发现其学生伊赛·舒尔的天才，这无疑是他的功劳。弗罗贝尼乌斯的有限群表示论后来在量子力学和理论物理中找到了重要应用，这可能并不完全令这位对数学持有如此“纯粹”观点的人感到高兴。

在弗罗贝尼乌斯职业生涯末期所研究的课题中，有正矩阵和非负矩阵。他引入了矩阵不可约性的概念，他在1910年前后撰写的包含这一理论的论文至今仍是该学科的基本成果。弗罗贝尼乌斯的许多论文读起来像他研究课题的当代教科书，这一事实清楚地表明，他在许多不同领域的工作对塑造今天所研究的数学具有何等重要性。话虽如此，他也确实为已经存在的领域做出了基础性贡献，而他并没有像一些最伟大的数学家那样开创任何全新的数学领域。

在5中，Haubrich对弗罗贝尼乌斯的工作给出了如下概述：-

他数学实践中最引人注目的方面是他在计算方面的非凡技巧。事实上，弗罗贝尼乌斯试图在很大程度上通过计算、代数的方法来解决数学问题。甚至他的分析工作也由代数和线性代数方法所引导。对于弗罗贝尼乌斯来说，概念性论证扮演了某种次要的角色。尽管他在相对抽象的框架中进行论证，但抽象本身并不是目的。在他看来，抽象的优点似乎主要在于它能带来更大的清晰性和精确性。

Georg Frobenius's father was Christian Ferdinand Frobenius, a Protestant parson, and his mother was Christine Elizabeth Friedrich. Georg was born in Charlottenburg which was a district of Berlin which was not incorporated into the city until 1920. He entered the Joachimsthal Gymnasium in 1860 when he was nearly eleven years old and graduated from the school in 1867. In this same year he went to the University of Göttingen where he began his university studies but he only studied there for one semester before returning to Berlin.

Back at the University of Berlin he attended lectures by Kronecker, Kummer and Weierstrass. He continued to study there for his doctorate, attending the seminars of Kummer and Weierstrass, and he received his doctorate (awarded with distinction) in 1870 supervised by Weierstrass. In 1874, after having taught at secondary school level first at the Joachimsthal Gymnasium then at the Sophienrealschule, he was appointed to the University of Berlin as an extraordinary professor of mathematics.

For the description of Frobenius's career so far, the attentive reader may have noticed that no mention has been made of him receiving his habilitation before being appointed to a teaching position. This is not an omission, rather it is surprising given the strictness of the German system that this was allowed. Details of this appointment are given in [3] but we should say that it must ultimately have been made possible due to strong support from Weierstrass who was extremely influential and considered Frobenius one of his most gifted students.

Frobenius was only in Berlin for a year before he went to Zürich to take up an appointment as an ordinary professor at the Eidgenössische Polytechnikum. For seventeen years, between 1875 and 1892, Frobenius worked in Zürich. He married there and brought up a family and did much important work in widely differing areas of mathematics. We shall discuss some of the topics which he worked on below, but for the moment we shall continue to describe how Frobenius's career developed.

In the last days of December 1891 Kronecker died and, therefore, his chair in Berlin became vacant. Weierstrass, strongly believing that Frobenius was the right person to keep Berlin in the forefront of mathematics, used his considerable influence to have Frobenius appointed. However, for reasons which we shall discuss in a moment, Frobenius turned out to be something of a mixed blessing for mathematics at the University of Berlin.

The positive side of his appointment was undoubtedly his remarkable contributions to the representation theory of groups, in particular his development of character theory, and his position as one of the leading mathematicians of his day. The negative side came about largely through his personality which is described in [5] as:-

... occasionally choleric, quarrelsome, and given to invectives.

Biermann, in [3], looks more closely at his character (no pun intended!), and how it affected the success of mathematical education at the university. He describes the strained relationships which developed between Frobenius and his colleagues at Berlin. He had such high standards that in the end these did not serve Berlin well. He [3]:-

... suspected at every opportunity a tendency of the Ministry to lower the standards at the University of Berlin, in the words of Frobenius, to the rank of a technical school ... Even so, Fuchs and Schwarz yielded to him, and later Schottky, who was indebted to him alone for his call to Berlin. Frobenius was the leading figure, on whom the fortunes of mathematics at Berlin university rested for 25 years. Of course, it did not escape him, that the number of doctorates, habilitation, and docents slowly but surely fell off, although the number of students increased considerably. That he could not prevent this, that he could not reach his goal of maintaining unchanged the times of Weierstrass, Kummer and Kronecker also in their external appearances, but to witness helplessly these developments, was doubly intolerable for him, with his choleric disposition.

We should not be too hard on Frobenius for, as Haubrich explains in [5], Frobenius's attitude was one which was typical of all professors of mathematics at Berlin at this time:-

They all felt deeply obliged to carry on the Prussian neo-humanistic tradition of university research and teaching as they themselves had experienced it as students. This is especially true of Frobenius. He considered himself to be a scholar whose duty it was to contribute to the knowledge of pure mathematics. Applied mathematics, in his opinion, belonged to the technical colleges.

The view of mathematics at the University of Göttingen was, however, very different. This was a time when there was competition between mathematicians in the University of Berlin and in the University of Göttingen, but it was a competition that Göttingen won, for there mathematics flourished under Klein, much to Frobenius's annoyance. In [3] Biermann writes that:-

The aversion of Frobenius to Klein and S Lie knew no limits ...

Frobenius hated the style of mathematics which Göttingen represented. It was a new approach which represented a marked change from the traditional style of German universities. Frobenius, as we said above, had extremely traditional views. In a letter to Hurwitz, who was a product of the Göttingen system, he wrote on 3 February 1896 (see [4]):-

If you were emerging from a school, in which one amuses oneself more with rosy images than hard ideas, and if, to my joy, you are also gradually becoming emancipated from that, then old loves don't rust. Please take this joke facetiously.

One should put the other side of the picture, however, for in [9] Siegel, who knew Frobenius for two years from 1915 when he became a student until Frobenius's death, relates his impression of Frobenius as having a warm personality and expresses his appreciation of his fast-paced varied and deep lectures. Others would describe his lectures as solid but not stimulating.

To gain an impression of the quality of Frobenius's work before the time of his appointment to Berlin in 1892 we can do no better than to examine the recommendations of Weierstrass and Fuchs when Frobenius was elected to the Berlin Academy of Sciences in 1892. Fairly extensive quotes from this document, and another similar document from Fuchs and Helmholtz, are given in [4] but we quote a short extract to show the power, variety and high quality of Frobenius's work in his Zürich years. Weierstrass and Fuchs list 15 topics on which Frobenius had made major contributions:-

1. On the development of analytic functions in series.
2. On the algebraic solution of equations, whose coefficients are rational functions of one variable.
3. The theory of linear differential equations.
4. On Pfaff's problem.
5. Linear forms with integer coefficients.
6. On linear substitutions and bilinear forms...
7. On adjoint linear differential operators...
8. The theory of elliptic and Jacobi functions...
9. On the relations among the 28 double tangents to a plane of degree 4.
10. On Sylow's theorem.
11. On double cosets arising from two finite groups.
12. On Jacobi's covariants...
13. On Jacobi functions in three variables.
14. The theory of biquadratic forms.
15. On the theory of surfaces with a differential parameter.

In his work in group theory, Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups. He published Über Gruppen von vertauschbaren Elementen Ⓣ in 1879 (jointly with Stickelberger, a colleague at Zürich) which looks at permutable elements in groups. This paper also gives a proof of the structure theorem for finitely generated abelian groups. In 1884 he published his next paper on finite groups in which he proved Sylow's theorems for abstract groups (Sylow had proved his theorem as a result about permutation groups in his original paper). The proof which Frobenius gives is the one, based on conjugacy classes, still used today in most undergraduate courses.

In his next paper in 1887 Frobenius continued his investigation of conjugacy classes in groups which would prove important in his later work on characters. In the introduction to this paper he explains how he became interested in abstract groups, and this was through a study of one of Kronecker's papers. It was in the year 1896, however, when Frobenius was professor at Berlin that his really important work on groups began to appear. In that year he published five papers on group theory and one of them Über die Gruppencharactere on group characters is of fundamental importance. He wrote in this paper:-

I shall develop the concept [of character for arbitrary finite groups] here in the belief that through its introduction, group theory will be substantially enriched.

This paper on group characters was presented to the Berlin Academy on July 16 1896 and it contains work which Frobenius had undertaken in the preceding few months. In a series of letters to Dedekind, the first on 12 April 1896, his ideas on group characters quickly developed. Ideas from a paper by Dedekind in 1885 made an important contribution and Frobenius was able to construct a complete set of representations by complex numbers. It is worth noting, however, that although we think today of Frobenius's paper on group characters as a fundamental work on representations of groups, Frobenius in fact introduced group characters in this work without any reference to representations. In was not until the following year that representations of groups began to enter the picture, and again it was a concept due to Frobenius. Hence 1897 is the year in which the representation theory of groups was born.

Over the years 1897-1899 Frobenius published two papers on group representations, one on induced characters, and one on tensor product of characters. In 1898 he introduced the notion of induced representations and the Frobenius Reciprocity Theorem. It was a burst of activity which set up the foundations of the whole of the machinery of representation theory.

In a letter to Dedekind on 26 April 1896 Frobenius gave the irreducible characters for the alternating groups $A_{4}, A_{5}$, the symmetric groups $S_{4}, S_{5}$ and the group $PSL(2,7)$ of order 168. He completely determined the characters of symmetric groups in 1900 and of characters of alternating groups in 1901, publishing definitive papers on each. He continued his applications of character theory in papers of 1900 and 1901 which studied the structure of Frobenius groups.

Only in 1897 did Frobenius learn of Molin's work which he described in a letter to Dedekind as "very beautiful but difficult". He reformulated Molin's work in terms of matrices and then showed that his characters are the traces of the irreducible representations. This work was published in 1897. Frobenius's character theory was used with great effect by Burnside and was beautifully written up in Burnside's 1911 edition of his Theory of Groups of Finite Order.

Frobenius had a number of doctoral students who made important contributions to mathematics. These included Edmund Landau who was awarded his doctorate in 1899, Issai Schur who was awarded his doctorate in 1901, and Robert Remak who was awarded his doctorate in 1910. Frobenius collaborated with Schur in representation theory of groups and character theory of groups. It is certainly to Frobenius's credit that he so quickly spotted the genius of his student Schur. Frobenius's representation theory for finite groups was later to find important applications in quantum mechanics and theoretical physics which may not have entirely pleased the man who had such "pure" views about mathematics.

Among the topics which Frobenius studied towards the end of his career were positive and non-negative matrices. He introduced the concept of irreducibility for matrices and the papers which he wrote containing this theory around 1910 remain today the fundamental results in the discipline. The fact so many of Frobenius's papers read like present day text-books on the topics which he studied is a clear indication of the importance that his work, in many different areas, has had in shaping the mathematics which is studied today. Having said that, it is also true that he made fundamental contributions to fields which had already come into existence and he did not introduce any totally new mathematical areas as some of the greatest mathematicians have done.

In [5] Haubrich gives the following overview of Frobenius's work:-

The most striking aspect of his mathematical practice is his extraordinary skill at calculations. In fact, Frobenius tried to solve mathematical problems to a large extent by means of a calculative, algebraic approach. Even his analytical work was guided by algebraic and linear algebraic methods. For Frobenius, conceptual argumentation played a somewhat secondary role. Although he argued in a comparatively abstract setting, abstraction was not an end in itself. Its advantages to him seemed to lie primarily in the fact that it can lead to much greater clearness and precision.