Schur

The authors are grateful to F.Schur, W.B.Klemmt, Alexei Kuntewich and V.Vengrinowich for helpful discussions. 

F. Schur, ]. Rohman, T.W. Guettinger Corrosion Detection. A new Approach Using Eddy Currents. ATA 1995 NDT FORUM Hartford, Connecticut, USA September 26-28, 1995. 

Laub, A.J. (1979) A Schur Method for Solving Riccati Equations, IEEE Trans, on Automat. Contr., AC-24, pp. 913-921. 

The polynomial (1.5) which I called cycle index is, if H is the symmetric group, equal to the principal character of H in representation theory. Professor Schur informed me that the cycle index of an arbitrary permutation group being really a subgroup of a symmetric group is of importance for the representation of this symmetric group. We will, however, not expand on the relationship between representation theory and our subject. 

I. Schur, Darstellungstheorie der Gruppen, Lecture Notes, Swiss Federal Institute of Technology, E. Stiefel, ed. (Zurich, 1936) (Cf. pp. 59-60). 

Professor Schur also made me aware of a consideration by Frobenius which is closely related to the argument given in Sec. 19. 

Sitzungsbericht der Akademie Berlin (1897), pp. 1152-1156. See formula (12) which holds for arbitrary finite groups of linear substitutions, not only for permutation groups. I m obliged to Prof. Schur for this reference. 

It was shown in [ReaR68] that the operation N A B) which is required by the superposition theorem is particularly simple if the operands A and B are the symmetric functions known as -functions (or Schur functions). In fact, for any two -functions X and /z ... 

WhiD80 White, D. E. A Polya interpretation of the Schur function. J. Comb. Theory A 28 (1980) 272-281. 

D Cysto-Myacyne O.W.G. (Schur)-comb. wfm Euvernil (Hcyden) wfm... 

Caldwell J, Ruddy S, Schur P, Austen K Acquired Cl inhibitor deficiency in lymphosarcoma. Clin Immunol Immunopathol 1972 1 39-52. 

Lafer EM, Valle RP, Moller A, Nordheim A, Schur PH, Rich A, Stollar BD (1983) J Clin Invest 71 314... 

Nelson, R.W. J.A. Schur. (1980). Assessment of effectiveness of geologic oscillation systems PATHS groundwater hydrologic model. Battelle, Pacific Northwest Laboratory, Richland, WA. 

If the matrix Q is positive semidefinite (positive definite) when projected into the null space of the active constraints, then (3-98) is (strictly) convex and the QP is a global (and unique) minimum. Otherwise, local solutions exist for (3-98), and more extensive global optimization methods are needed to obtain the global solution. Like LPs, convex QPs can be solved in a finite number of steps. However, as seen in Fig. 3-57, these optimal solutions can lie on a vertex, on a constraint boundary, or in the interior. A number of active set strategies have been created that solve the KKT conditions of the QP and incorporate efficient updates of active constraints. Popular methods include null space algorithms, range space methods, and Schur complement methods. As with LPs, QP problems can also be solved with interior point methods [see Wright (1996)]. 

There are two theorems of fundamental importance, known as Schur s lemmas, which are useful in the study of the irreducible representations of a group. 

Therefore by Schur s second lemma it follows that M = O. Taking the (k, s) element of (10), gives... 

From Schur s first lemma it follows that N must be a constant matrix, N = aE (say), where E is the unit matrix of order lt. Again, taking the (k, s)th element of (13) one gets... 

There are other important classes of symmetric functions, such as Schur functions. Jack polynomials, Macdonald polynomials, etc (see [54]). In [70], we shall discuss a geometric interpretation of Jack polynomials. 

In Section 6.1 we define irreducible representations. Then we state, prove and illustrate Schur s lemma. Schur s lemma is the statement of the all-or-nothing personality of irreducible representations. In the Section 6.2 we discuss the physical importance of irreducible representations. In Section 6.3 we introduce invariant integration and apply it to show that characters of irreducible representations form an orthonormal set. In the optional Section 6.4 we use the technology we have developed to show that finite-dimensional unitary representations are no more than the sum of their irreducible parts. The remainder of the chapter is devoted to classifying the irreducible representations of 5(7(2) and 50(3). 

In this section we will use the idea of invariant subspaces of a representation (see Definition 5.1) to define irreducible representations. Then we will prove Schur s lemma, which tells us that irreducible representations are indeed good building blocks. 

Proposition 6.2 (Schur s lemma) Suppose (G, Vi, pi) anJ (G, V2, pi) are irreducible representations of the same group G. Suppose that T Vi V2 is a homomorphism of representations. Then there are only two possible cases ... 

The next proposition says that there are no interesting homomorphisms from an irreducible representation to itself. We will use this consequence of Schur s lemma in our first prediction for the hydrogen atom. Proposition 7.7. For the statement of the proposition, some terminology is convenient. 

The following consequence of Schur s lemma will be useful in the proof that every polynomial restricted to the two-sphere is equal to a harmonic polynomial restricted to the two-sphere (Proposition 7.3). The idea is that once we decompose a representation into a Cartesian sum of irreducibles, every irreducible subrepresentation appears as a term in the sum. 

Proof. By Proposition 3.5, since V2 is finite dimensional we know that there is an orthogonal projection 112 with range V2. Because p is unitary, the linear transformation 112 is a homomorphism of representations by Proposition 5.4. Thus by Exercise 5.15 the restriction of 112 to Vi is a homomorphism of representations. By hypothesis, this homomorphism cannot be injective. Hence Schur s lemma (Proposition 6.2) implies that since Vi is irreducible, fl2[Vi] is the trivial subspace. In other words, Vi is perpendicidar to V2. ... 

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