Schur continued the work of his teacher Frobenius with many important works for
group theory and
representation theory. In addition, he published important results and elegant proofs of known results in almost all branches of classical algebra and number theory. His collected works are proof of this. There, his work on the theory of integral equations and infinite series can be found.
Linear groups In his doctoral thesis
Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen Issai Schur determined the polynomial representations of the general linear group GL(n, \mathbb{C}) on the field \mathbb{C} of
complex numbers. The results and methods of this work are still relevant today. In his book,
J.A. Green determined the polynomial representations of GL (n, \mathbb{K}) over infinite fields \mathbb{K} with arbitrary characteristic. It is mainly based on Schur's dissertation. Green writes, "This remarkable work (of Schur) contained many very original ideas, developed with superb algebraic skill. Schur showed that these (polynomial) representations are completely reducible, that each irreducible one is "homogeneous" of some degree r \geq 0, and that the equivalence types of irreducible polynomial representations of GL_n(\mathbb{C}), of fixed homogeneous degree r, are in one-one correspondence with the partitions \lambda = (\lambda_1, \ldots, \lambda_n) of r into not more than n parts. Moreover Schur showed that the character of an irreducible representation of type \lambda is given by a certain symmetric function {\underline{S}}_{\lambda} in n variables (since described as a "Schur function")." According to Green, the methods of Schur's dissertation today are important for the theory of algebraic groups. In 1927 Schur, in his work
On the rational representations of the general linear group, gave new proofs for the main results of his dissertation. If V is the natural n-dimensional \mathbb{C} vector space on which GL(n, \mathbb{C}) operates, and if r is a natural number, then the r-fold tensor product V^{\otimes r} over \mathbb{C} is a GL(n, \mathbb {C})-module, on which the symmetric group S_r of degree r also operates by permutation of the tensor factors of each generator v_1 \otimes \ldots \otimes v_r of V^{\otimes r}. By exploiting these S_r - GL(n, \mathbb{C})-bimodule actions on V^{\otimes r}, Schur manages to find elegant proofs of his sentences. This work of Schur was once very well known.
Professorship in Berlin In Berlin, Schur was a highly respected member of the academic world, an apolitical scholar. As a leading mathematician and a successful teacher, he held a prestigious chair at the University of Berlin for 16 years. Until 1933, his research group had an excellent reputation. His faculty worked with representation theory, which was extended by his students (including solvable groups, combinatorics, and matrix theory). Schur made fundamental contributions to algebra and group theory which, according to
Hermann Weyl, were comparable in scope and depth to those of
Emmy Noether (1882–1935). When Schur's lectures were canceled in 1933, there was an outcry among the students and professors who appreciated him and liked him. By the efforts of his colleague
Erhard Schmidt Schur was allowed to continue lecturing until the end of September 1935. Schur was the last Jewish professor who lost his job.
Zurich lecture In Switzerland, Schur's colleagues Heinz Hopf and George Pólya were informed of the dismissal of Schur in 1935. They tried to help as best they could. On behalf of the Mathematical Seminars chief
Michel Plancherel, on 12 December 1935 the school board president Arthur Rohn invited Schur to
une série de conférences sur la théorie de la représentation des groupes finis. At the same time he asked that the formal invitation should come from President Rohn, ''comme le prof. Schur doit obtenir l'autorisation du ministère compétent de donner ces conférences''. George Pólya arranged from this invitation of the Mathematical Seminars the Conference of the Department of Mathematics and Physics on 16 December. Meanwhile, on 14 December the official invitation letter from President Rohn had already been dispatched to Schur. Schur was promised for his guest lecture a fee of CHF 500. Schur did not reply until 28 January 1936, on which day he was first in the possession of the required approval of the local authority. He declared himself willing to accept the invitation. He envisaged beginning the lecture on 4 February. Schur spent most of the month of February in Switzerland. Before his return to Germany he visited his daughter in Bern for a few days, and on 27 February he returned via Karlsruhe, where his sister lived, to Berlin. In a letter to Pólya from Berne, he concludes with the words:
From Switzerland I take farewell with a heavy heart. In Berlin, meanwhile, mathematician and Nazi Ludwig Bieberbach, in a letter dated 20 February 1936, informed the Reich Minister for Science, Art, and Education on the journey of Schur, and announced that he wanted to find out what was the content of the lecture in Zurich. ==Significant students==