Issai Schur

How to use this course

Issai Schur’s name is attached to more results than almost any other mathematician of his generation: a lemma, a multiplier, a decomposition, a complement, a product, polynomials, an index, a test, a theorem in Ramsey theory and an algorithm in function theory. Most students meet several of them before they know they come from one person. This course gathers them, starting from the representation theory he built with his teacher Georg Frobenius, and tells the story of a career that ended in dismissal and exile.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 to 07 form a short course in the representation theory of finite groups and of \(\mathrm{GL}_n\); Lessons 08 to 10 are independent and can be read in any order.

Timeline

Mogilev and Libau, 1875–1894

Issai Schur was born on 10 January 1875 in Mogilev, a town on the Dnieper in the western Russian Empire, into a Jewish family. His father Moses Schur was a merchant; his mother was Golde Schur, née Landau. His given name appears in early records as Schaia; he used the form Issai from about 1901. Some of his papers were printed under “J. Schur”, a source of confusion in later bibliographies.

At thirteen he moved to Libau, a Baltic port in the province of Courland (now Liepāja in Latvia), where an older brother and sister lived. He attended the German-language Nicolai Gymnasium from 1888 to 1894 and graduated at the top of his class with a gold medal. The school gave him the German he would write in for the rest of his life, and a route to a German university.

Berlin and Frobenius, 1894–1913

In October 1894 Schur entered the University of Berlin to study mathematics and physics. Berlin mathematics was then led by Georg Frobenius, Lazarus Fuchs and Hermann Amandus Schwarz. Frobenius had just created the theory of group characters (1896), and Schur became his most important student.

The thesis

Schur’s dissertation, Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen (1901), received the grade summa cum laude. It solved a problem far beyond finite groups: determine all polynomial representations of the general linear group \(\mathrm{GL}_n(\mathbb C)\). Schur showed they are completely reducible, classified the irreducible ones by partitions, and computed their characters, the functions now called Schur polynomials (Lesson 07).

He completed his habilitation in 1903 and became a Privatdozent in Berlin. The next eight years were extraordinarily productive. In 1904 he created the theory of projective representations (Lesson 06); in 1905 he rebuilt character theory on the lemma that bears his name (Lesson 04); in 1906 he and Frobenius classified real representations (Lesson 05); in 1909 and 1911 came results on matrices that are now in every linear algebra text (Lesson 08). In 1906 he married Regina Frumkin, a physician. They had two children, Georg and Hilde.

Bonn and the return, 1913–1922

In 1913 Schur became extraordinary professor at Bonn, succeeding Felix Hausdorff. Frobenius wanted him back. In 1916, after pressure from Frobenius, Schur returned to Berlin as an extraordinary professor, succeeding Johannes Knoblauch. When Frobenius died in 1917, the faculty considered Schur and Constantin Carathéodory for the chair, and chose Carathéodory.

Recognition followed quickly nonetheless. In 1919 Schur received a personal chair and was elected to the Leopoldina. In 1921 he took over the chair of the retiring Friedrich Schottky, and in 1922 he was elected to the Prussian Academy of Sciences on the proposal of Max Planck. In 1918 he had co-founded the journal Mathematische Zeitschrift, which he helped edit until 1936.

These were also years of new directions. The sum-free theorem of 1916, a founding result of Ramsey theory, came from a question about Fermat’s equation modulo primes (Lesson 09). The papers of 1917 and 1918 on bounded analytic functions launched a line of work that runs to modern signal processing (Lesson 10).

Schur’s lemma and orthogonality

For the general reader. A representation lets a group act by matrices. Frobenius found the characters of these representations using elaborate determinant calculations. In 1905 Schur showed that almost everything follows from one simple observation: a linear map that respects two indecomposable symmetries is either zero or an isomorphism. This lemma is now the first theorem of representation theory.

Proof. \(\ker T\) is a \(G\)-stable subspace of \(V\) and \(\operatorname{im}T\) a \(G\)-stable subspace of \(W\). Irreducibility forces each to be trivial or everything. For the second part, let \(\lambda\) be an eigenvalue of \(T\); then \(T-\lambda\) commutes with \(\rho\) and is not invertible, so it is zero.

Orthogonality

Let \(G\) be finite and \(k=\mathbb C\). For any linear map \(A:V\to W\), the average \(\tilde A=\frac1{|G|}\sum_g\sigma(g)A\rho(g)^{-1}\) intertwines \(\rho\) and \(\sigma\). Schur’s lemma gives \(\tilde A=0\) if \(\rho\not\cong\sigma\), and \(\tilde A=\frac{\operatorname{tr}A}{\dim V}\mathrm{id}\) if \(\rho=\sigma\). Taking \(A\) to be matrix units yields:

Hence a representation is determined by its character, the multiplicity of an irreducible \(\chi\) in \(V\) is \(\langle\chi_V,\chi\rangle\), and \(V\) is irreducible exactly when \(\langle\chi_V,\chi_V\rangle=1\). The same argument with Haar measure in place of averaging works for compact groups, as Hermann Weyl showed in the 1920s.

Real, complex, quaternionic: indicator and index

For the general reader. A representation defined with complex numbers might or might not be rewritable with real numbers alone, or with rational numbers. Frobenius and Schur found a one-line test for the real case, and Schur introduced an integer, now called the Schur index, that measures how far a representation is from being definable over a given field.

Sketch. Write \(\chi(g^2)=\chi_{S^2V}(g)-\chi_{\Lambda^2V}(g)\). So \(\nu(\chi)\) is the number of invariant symmetric bilinear forms minus the number of invariant alternating forms on \(V\). An invariant non-zero bilinear form is a \(G\)-isomorphism \(V\to V^*\), which exists only when \(\chi=\bar\chi\), and by Schur’s lemma it is unique up to scalar and either symmetric or alternating. A symmetric invariant form, combined with an invariant Hermitian form, gives a real structure; an alternating one gives a quaternionic structure.

Summing over all irreducibles gives a counting formula:

The Schur index

For a field \(K\subseteq\mathbb C\) containing the values of \(\chi\), the Schur index \(m_K(\chi)\) is the smallest degree \([L:K]\) of an extension over which the representation can be realized. In 1906 Schur proved that \(m_{\mathbb Q}(\chi)\) divides \(\chi(1)\). The index connects representation theory to the theory of division algebras of Lesson 10 of the Noether course: it is the index of a central simple algebra appearing in the group algebra \(K[G]\).

Projective representations and the multiplier

For the general reader. In quantum mechanics a state is a vector only up to a phase, so symmetries act by matrices that compose correctly only up to a scalar. Schur studied such “projective” representations in 1904, decades before physics needed them. He showed that each finite group has a finite abelian group, the multiplier, measuring the obstruction, and a covering group on which every projective representation becomes an ordinary one.

A projective representation is a map \(\rho:G\to\mathrm{GL}_n(\mathbb C)\) with \(\rho(g)\rho(h)=\alpha(g,h)\rho(gh)\) for scalars \(\alpha(g,h)\in\mathbb C^\times\). Associativity forces the cocycle identity \(\alpha(g,h)\alpha(gh,k)=\alpha(h,k)\alpha(g,hk)\). Rescaling \(\rho(g)\) by \(c(g)\) changes \(\alpha\) by a coboundary \(c(g)c(h)/c(gh)\).

If \(G=F/R\) with \(F\) free, then

Schur worked with covering groups built from presentations; the formula in this clean form was found by Heinz Hopf in 1942, as a statement about the second homology of a space with fundamental group \(G\), and became one of the starting points of group cohomology. Schur also showed that the exponent of \(M(G)\) divides \(|G|\). A by-product of the same papers is Schur’s theorem that if the centre \(Z(G)\) of a group has finite index, then the commutator subgroup \([G,G]\) is finite.

Spin representations

In 1911 Schur computed \(M(S_n)=\mathbb Z/2\) for \(n\ge4\) and found all projective representations of the symmetric and alternating groups. Their characters are governed by the Schur \(Q\)-functions. The double cover of the rotation group \(\mathrm{SO}(3)\) by \(\mathrm{SU}(2)\), which describes electron spin, is the continuous analogue.

Schur functions and Schur–Weyl duality

For the general reader. Schur’s thesis asked which matrix-valued polynomials \(A\mapsto\rho(A)\) respect multiplication. The answer is indexed by partitions, the ways of writing a whole number as a sum, and the characters are symmetric polynomials now called Schur polynomials. They link the symmetric group, the general linear group and combinatorics, and appear today in algebraic geometry, probability and physics.

A partition \(\lambda=(\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n\ge0)\) of \(k=|\lambda|\) with at most \(n\) parts defines

where the sum runs over semistandard Young tableaux of shape \(\lambda\) with entries in \(\{1,\dots,n\}\) (weakly increasing along rows, strictly down columns), and \(x^T=\prod_i x_i^{\#\{i\text{ in }T\}}\). The denominator is the Vandermonde determinant.

Weyl, who used the result throughout The Classical Groups (1939), gave it wide currency, hence the double name. Taking characters gives the Frobenius formula \((x_1+\cdots+x_n)^k=\sum_\lambda f^\lambda s_\lambda(x)\) with \(f^\lambda=\dim S^\lambda\).

Matrices: triangular form, products, majorization

For the general reader. Several of the most used facts in matrix theory are Schur’s. Every square matrix can be put in triangular form by a rotation of coordinates. The entrywise product of two positive semidefinite matrices is again positive semidefinite. The diagonal of a symmetric matrix is always “more even” than its eigenvalues. Each began as a lemma in a paper on something else.

Proof. Pick a unit eigenvector \(u_1\), extend to an orthonormal basis, and induct on the lower-right \((n-1)\times(n-1)\) block. The Frobenius norm is unitarily invariant, so \(\sum|a_{ij}|^2=\sum|t_{ij}|^2=\sum|\lambda_i|^2+\sum_{i<j}|t_{ij}|^2\). Equality means \(T\) is diagonal, i.e. \(A\) is unitarily diagonalizable, i.e. normal.

Proof. Write \(B=\sum_k v_kv_k^*\). Then \(A\circ B=\sum_k D_kAD_k^*\) with \(D_k=\operatorname{diag}(v_k)\), a sum of positive semidefinite matrices.

Alfred Horn proved the converse in 1954. Functions that respect majorization are called Schur-convex. Finally, in his 1917 work on bounded functions Schur used the determinant formula for a block matrix with \(A\) invertible,

and \(D-CA^{-1}B\) was named the Schur complement by Emilie Haynsworth in 1968.

Sum-free sets and Fermat modulo p

For the general reader. Colour the numbers 1, 2, 3, … with a fixed number of colours, however cleverly. Schur proved in 1916 that some colour will always contain three numbers with \(x+y=z\). He wanted this to show that Fermat’s equation always has solutions modulo large primes. His theorem was one of the first results of what is now called Ramsey theory.

Proof

Let \(N\) be large and colour \(\{1,\dots,N\}\). Colour each edge \(\{i,j\}\) of the complete graph on vertices \(0,1,\dots,N\) by the colour of \(|i-j|\). Ramsey’s theorem for graphs (proved in 1930, but Schur’s argument is the same pigeonhole induction) gives, once \(N+1\ge R_c(3)\), where \(R_c(3)\) is the least number of vertices forcing a monochromatic triangle in any \(c\)-colouring of the edges, a monochromatic triangle \(i<j<k\). Then \(x=j-i\), \(y=k-j\), \(z=k-i\) share a colour and \(x+y=z\). So \(S(c)\le R_c(3)-1\). The pigeonhole recursion \(R_c(3)\le c\,(R_{c-1}(3)-1)+2\) with \(R_1(3)=3\) gives \(R_c(3)\le\lfloor e\,c!\rfloor+1\).

Proof. The \(m\)-th powers form a subgroup \(H\) of \((\mathbb Z/p)^\times\) of index \(d=\gcd(m,p-1)\le m\). Colour \(1,\dots,p-1\) by their coset of \(H\). Since \(p-1\ge S(m)\ge S(d)\), there are \(a,b,c\) in one coset \(tH\) with \(a+b=c\). Write \(a=tx^m\), \(b=ty^m\), \(c=tz^m\) and cancel \(t\).

So Fermat’s Last Theorem cannot be proved by working modulo a single prime. The exact values known are \(S(1)=2\), \(S(2)=5\), \(S(3)=14\), \(S(4)=45\) and \(S(5)=161\), the last found by a computer proof by Marijn Heule in 2017. Schur’s student Richard Rado generalized the theorem to all linear equations (Rado’s theorem, 1933).

Bounded analytic functions

For the general reader. Which power series define functions bounded by 1 on the unit disc? Schur gave an answer in 1917 by an algorithm that peels off one coefficient at a time, like a continued fraction. The numbers it produces, the Schur parameters, appear today in control theory, filter design and the theory of orthogonal polynomials.

Let \(\mathcal S\) be the class of analytic \(f\) on \(\mathbb D=\{|z|<1\}\) with \(|f|\le1\). For \(f\in\mathcal S\) set \(\gamma_0=f(0)\). If \(|\gamma_0|<1\) define

By the Schwarz lemma \(f_1\in\mathcal S\). Repeating gives \(f_2,f_3,\dots\) and parameters \(\gamma_n=f_n(0)\).

Schur also expressed the condition through determinants built from the Taylor coefficients, and it is in this work that he used the block determinant formula of Lesson 08. Arthur Cohn adapted the method in 1922 into the Schur–Cohn test for whether a polynomial has all its zeros in the disc, a standard stability criterion for discrete-time systems.

Schur worked in analysis in other ways too. His test (1911) bounds integral operators: if \(K\ge0\) and positive functions \(p,q\) satisfy \(\int K(x,y)q(y)\,dy\le\alpha p(x)\) and \(\int K(x,y)p(x)\,dx\le\beta q(y)\), then the operator with kernel \(K\) has \(L^2\) norm at most \(\sqrt{\alpha\beta}\). In 1921 he showed that in \(\ell^1\) weakly convergent sequences converge in norm, the first example of what is now called the Schur property.

The Berlin school, 1916–1933

From his return in 1916 until 1933 Schur was the centre of algebra in Berlin. His lectures were famous for their clarity and drew large audiences; Walter Ledermann, who attended from 1928, later called them “perfect in form and content”. Schur set problems in his seminar and sometimes in print. In 1921 one of them was solved simultaneously by the brothers Richard and Alfred Brauer and by Heinz Hopf.

He supervised around two dozen doctorates. Among his students were Heinz Prüfer (1921), Félix Pollaczek (1922), Dora Prölß (1922, the first woman to take a mathematics doctorate in Berlin), Karl Dörge (1925), Richard Brauer (1926), Isaac Schoenberg (1926), Alfred Brauer (1928), Bernhard Neumann (1932), Wilhelm Specht (1932), Richard Rado (1933) and Helmut Wielandt (1935). Robert Frucht, Rose Peltesohn and Wolfgang Hahn also completed theses with him, and Kurt Hirsch, Hanna Neumann and Max Schiffer studied under him. Richard Brauer carried his representation theory into modular characters and the classification of finite simple groups.

Schur had a long friendship with George Pólya, with whom he shared interests in inequalities and analysis. Hermann Weyl later placed the breadth and depth of Schur’s work alongside that of Emmy Noether, though their styles were opposite: Schur preferred explicit, concrete methods to Noether’s abstraction.

Dismissal, exile and legacy, 1933–1941

In 1933 the Nazi Law for the Restoration of the Professional Civil Service led to Schur being placed on leave. His colleague Erhard Schmidt fought for him, and from the winter semester 1933–34 he was allowed to give some specialist lectures again. He declined an invitation to the University of Wisconsin, feeling unable to start again in English. He was allowed to teach until the end of September 1935, when he was dismissed; he was among the last Jewish professors in Berlin to lose his post.

In February 1936 he gave a series of lectures on representation theory at ETH Zürich. Back in Berlin, Ludwig Bieberbach, a committed Nazi, objected to his continued presence in Academy commissions. In 1938 Schur resigned from them and then from the Academy itself. That summer the Gestapo summoned him; his wife went in his place. The family could not pay the emigration tax demanded of Jews until anonymous donors covered the shortfall. Early in 1939 Schur left for Bern, where his daughter lived, and then emigrated with his wife to Palestine.

He settled in Tel Aviv. He died of a heart attack on 10 January 1941, his 66th birthday. In 1942 the German authorities posthumously revoked his citizenship.

Legacy

• Representation theory: Schur’s lemma, orthogonality, the indicator and index, the multiplier and covering groups, spin representations of \(S_n\).
• Algebraic combinatorics: Schur polynomials and Schur–Weyl duality, central to symmetric functions, Schubert calculus and quantum groups.
• Linear algebra: the Schur decomposition, behind the QR algorithm for eigenvalues; the Schur product theorem; the Schur complement; majorization.
• Combinatorics and analysis: Schur’s theorem in Ramsey theory, Schur numbers, the Schur algorithm and the Schur–Cohn stability test.
• Works: his Gesammelte Abhandlungen, edited by Alfred Brauer and Hans Rohrbach, appeared in three volumes in 1973; his lectures on invariant theory were published in 1968.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Issai Schur, MacTutor History of Mathematics.
2. Issai Schur, Wikipedia.
3. J. Batt, Schur, Issai, Neue Deutsche Biographie 23 (2007).
4. R. Siegmund-Schultze, Issai Schur and his students, lecture slides, Bielefeld, 2013.
5. A. Joseph, A. Melnikov and R. Rentschler (eds.), Studies in Memory of Issai Schur, Birkhäuser, 2003.
6. W. Ledermann, Issai Schur and his school in Berlin, Bull. London Math. Soc. 15 (1983).

Mathematics

1. I. Schur, Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen, dissertation, Berlin, 1901.
2. I. Schur, Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. 127 (1904).
3. I. Schur, Neue Begründung der Theorie der Gruppencharaktere, Sitzungsber. Preuss. Akad. Wiss. (1905).
4. G. Frobenius and I. Schur, Über die reellen Darstellungen der endlichen Gruppen, Sitzungsber. Preuss. Akad. Wiss. (1906).
5. I. Schur, Arithmetische Untersuchungen über endliche Gruppen linearer Substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1906).
6. I. Schur, Über die charakteristischen Wurzeln einer linearen Substitution mit einer Anwendung auf die Theorie der Integralgleichungen, Math. Ann. 66 (1909).
7. I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math. 139 (1911).
8. I. Schur, Über die Kongruenz \(x^m+y^m\equiv z^m\pmod p\), Jahresber. Deutsch. Math.-Verein. 25 (1916).
9. I. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. reine angew. Math. 147 (1917), 148 (1918).
10. I. Schur, Über lineare Transformationen in der Theorie der unendlichen Reihen, J. reine angew. Math. 151 (1921).
11. I. Schur, Gesammelte Abhandlungen, 3 vols., ed. A. Brauer and H. Rohrbach, Springer, 1973.
12. W. Fulton, Young Tableaux, Cambridge University Press, 1997.
13. R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2013.
14. M. J. H. Heule, Schur Number Five, arXiv:1711.08076.