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Generalized FS×W versus Dyson’s classification of irreducible representations
1.G. Frobenius and I. Schur, “Uber die reellen Darstellungen der endlichen Gruppen,” Berl. Berichte, 186 (1906).
2.E. P. Wigner, Group Theory (Academic, New York, 1959), Sec. 24.
3.L. C. Biedenharn and J. D. Louck, “The Racah‐Wigner Algebra in Quantum Theory,” in Encyclopedia of Mathematics and its Applications, edited by G. C. Rota (Addison‐Wesley, Reading, MA, 1981), Chap. 5, Sees. 6 and 10.
4.R. Ascoli, C. Garola, L. Solombrino, and G. Teppati, “Real versus Complex Representations and Linear‐Antilinear Commutant,” in Physical Reality and Mathematical Description, edited by C. P. Enz and J. Mehra (Reidel, Dordrecht, 1974), Sec. 1. pp. 239–251.
5.See Ref. 2, Sec. 26.
6.F. J. Dyson, “The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics,” J. Math. Phys. 3, 1199 (1962), Sees. 3–5; see also Ref. 3, Chap. 5, Sec. 10.
7.H. Weyl, The Classical Groups, Their Invariants and Representations (Princeton U.P., Princeton, NJ, 1939), Chap. 3, Sec. 5, Theorem (3.5.B);
7.see also Ref. 6, Sec. 2.
8.See Ref. 6, Sec. 1.
9.See Ref. 4, Sec. 3, Table.
10.C. Garola and L. Solombrino, “Commutation in Vector Spaces over Division Rings with a Conjugation,” Linear Algebra Appl. 36, 41 (1981), Sec. 1, Table 1.
11.C. Garola and L. Solombrino, “Irreducible Linear‐Antilinear Representations and Internal Symmetries,” J. Math. Phys. 22, 1350 (1981), Sec. 3. 1
12.See Ref. 6, Sec. 4.
13.N. Bourbaki, Éléments de Mathématique, Algèbre, Chap. 2: Algèbre lineaire (Hermann, Paris, 1962), Sec. 8.
14.We note that, whenever X has finite dimension has dimension 2n, see R. Ascoli, C. Garola, L. Solombrino, and G. Teppati, “Vector Spaces over Fields with a Conjugation and Linear‐Antilinear Commutants,” Rend. Mat. Ser. VI 10 (1), 129 (1977), Sec. 2, Theorem 1.
15.See Ref. 4, Sec. 3, Lemma 1.
16.See Ref. 4, Sec. 3, Theorem 3.
17.See Ref. 11, Sees. 3 and 4, Propositions 1 and 2.
18.See Ref. 4, Sec. 4.
19.N. Bourbaki, Éléments de Mathématique, Algèbre (Hermann, Paris, 1973), Chap. 8; see in particular Sec. 4, No. 4, Propositions 4 and 5.
20.N. Bourbaki, Éléments de Mathématique, Algèbre (Hermann, Paris, 1971), Chap. 3, Sec. 1.
21.See Ref. 7, Chap. 3, Sec. 4.
22.Whenever K is any field with a conjugation and S any antilinear mapping the first statement in Lemma 1 can be deduced from Proposition 1 in Ref. 11, Sec. 3.
23.See Ref. 14, Sec. 5, Theorem 4.
24.See Ref. 11, Sec. 3, proof of Proposition 1.
25.We observe that the first statement in the theorem holds invariant whenever C is generalized to be any algebraically closed field K with a conjugation; in the proof, no change is needed besides the substitution
26.;Observe that the decomposition does not coincide with the one induced in by the decomposition of X introduced in the first part of the present proof.
27.An immediate proof of the first part of the second statement in the theorem can also be given as follows. Because of a statement proved by the authors in a previous paper (see Ref. 10, Sec. 3, Proposition 6) is potentially real iff has divisors of zero. By inspection of the possible forms of listed in Theorem 1, we see that this case occurs iff is reducible in
28.See Ref. 10, Sec. 2, first remark in the proof of Proposition 1.
29.See Ref. 7, Chap. 3, Sec. 1 and Ref. 6, Sec. 2, Remark 4.
30.See Ref. 10, Sec. 3, Proposition 5.
31.See Ref. 10, Sec. 2. Proposition 1.
32.See Ref. 10, Sec. 2, Proposition 4.
33.It follows in particular that Dyson’s cases CC1 and CC2 correspond to the cases in row 3, column 3 and to the upper subcase in row 1, column 3, of Table III, respectively; the distinction between the two cases is not irrelevant, contrary to our last sentence in Ref. 11, footnote 15.
34.These results complete, in some sense, the result in the remark 5 in Ref. 11, Sec. 5, which refer to columns 1, 2, 3 of Table III only. However, this remark has been stated improperly (its proof being correct) since no reference to the case U complex should have been made. The following constitutes a proper (and simplified) statement. Whenever is isomorphic to R or to C (columns 2 and 1,3, respectively, in Table II) and then the cases (a) potentially real and (b) pseudoreal, can be characterized by and for any and for any respectively.
35.See Ref. 6, last part of Sec. 4.
36.The equivalence between the two definitions is an immediate consequence of the equation which holds for any (see footnote 28).
37.See next section and Ref. 6, Sec. 5.
38.It is worthy of note that such a basis can easily be constructed whenever an involutory is known. Indeed, let us consider the mapping Then, some exists such that S is invertible. For, should this not be the case, for every some nonzero would exist such that that is, which is impossible because of basic statements about the eigenvalues of any linear mapping of X. Whenever S is invertible, we get hence Now, let us put Then, easily, i.e., Since it follows for every which proves that has the desired property.
39.We notice that, whenever is the matrix realization with respect to any basis in X of some the matrix realization of M with respect to the basis in is given by if M is a linear, and by if M is antilinear. In particular, we obtain and and [see also the proof of (iv)⇒(i) in Theorem 2]. We observe explicitly that, whenever M is antilinear, it is realized in X by the pair (see footnote 18), while the matrix alone realizes M in
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