Factorization lemma

In this expression the coefficients in brackets < > are the isoscalar factors (Clebsch-Gordan coefficients) for coupling two 0(4) and two 0(3) representations, respectively. They can be evaluated either analytically using Racah s factorization lemma (Section B.14) or numerically using subroutines explicitly written for this purpose.2... 

This celebrated factorization lemma (Wyboume, 1974 Biedenham and Louck, 1981) allows one to simplify considerably the calculations in the molecular case. The algebras of interest are... 

Since DN(UV-- -XY)Z = N(DUV- - -XY)Z, the theorem is proved for n + 1 factors. This lemma can be generalized by multiplying both sides of Eq. (10-196) by an arbitrary number of contracted factors, and using Eq. (10-195) to bring these factors within the N products. Wick s theorem now states that a T product can be decomposed into a unique sum of normal products as follows ... 

Composition series lead naturally to the notion of a composition factor, and, according to Lemma 4.2.4(i), composition factors must be simple. Thus, the above theorem on composition series gives reason to consider simple schemes as crucial in scheme theory. 

It follows immediately from Lemma 4.2.4(i) that composition factors are simple. 

Proof. The lemma immediately shows the equivalence of (1) and (2) and of (4) and (6). Clearly (5) implies (2), and (2) implies (3) because Homk(/t, fc) Hom((/4 k, k). RecaU from field theory now that a finite L over k has at most dimk(L) maps to k, and has exactly that number iff it is separable. But a map from A to k also kills all but one of the factors of A and vanishes on nilpotents in that one. Thus (3) is equivalent to (4). If they hold, then all maps A - k have separable image and thus actually map to k,. The kernels of the corresponding maps A k,->ks are primes, and (5) must hold since the number of these primes equals the dimension. ... 

Induction now shows that we can take any polynomial in the f with coefficient in (p0(A) and reduce it to have all exponents less than q. Hence A is a finitely generated module over B = cp0 A). This implies first of all that under A - B the dimension cannot go down. But since G is connected, A modulo its nilradical is a domain (6.6), and from (12.4) we see then that the kernel of k. Let M be the kernel, a maximal ideal of B. As B injects into A, we know BM injects into AM, and thus Am is a nontrivial finitely generated BM-module. By Nakayama s lemma then MAm Am, and so MA A. Any homomorphism x A- A/MA - fc then satisfies q>(x) = y. ... 

Lemma 8.65. (Adapted from [G0MR88].) On the factoring assumption (Definition 8.18), Construction 8.64 defines a strong and a weak claw-intractable family of permutation pairs. ... 

Lemma 9.19 (Summary of the factoring schemes). The factoring schemes from Definition 9.17 with the function rho have the following components that are actually used ... 

Lemma 9.22 (Complexity of the factoring schemes). In the following, the complexity of the schemes from Definition 9.17 and Lemma 9.19 is summarized. To avoid confusion when comparing the results with those of the discrete-logarithm scheme, where the size of k is usually different, the length I = r1 2 (=2k) is used as a parameter. 

Proof (Sketch). The idea is to redo everything in this section 4.9, up to this point, with etale in place of open immersion. The first difficulty which arises is that in the last paragraph of the proof of Lemma (4.9.2), the map i is now finite etale, making it necessary to know (4.9.2.S) for finite etale /, a fact given by Exercise (4.8.12)(b)(vi). The only other nontrivial modification is in the proof of (4.9.2.2), where the map X Xz X Y xzY should now be factored asXxzX W Y XzY with the first map an open immersion and the second proper, and then X should be defined to be the schematic image ofX—xz X. .. Q.E.D. 

Note that the realization functor — Iaj A SfwJT) A Shv(T) takes values in the fuU subcategory of simplicial sheaves of simplicial dimension zero, i.e. factors through a functor — a A ShifT) Sfw(Tj which is left adjoint to the restriction of C, to Shv(T). Together with Lemma 3.12 this fact can be used to obtain an alternative description of the homotopy category, 9(0 (T, I) as follows. 

The symbols a and y are multiplicity labels that are required if A and B occur more than once in the respective decompositions of A, x A2 and B, x B2. The symbol Pi is a multiplicity label of a different kind it is required when the reduction G yields more than one identical irreducible representation B,- in a given A,. Like the proof of the Wigner-Eckart theorem, the derivation of (51) depends crucially on Schur s lemma. The second factor on the right-hand side of eq. (50), for want of a better expression, has become known over the years as an isoscalar factor, in analogy to the situation for SU(3) (Edmonds 1962). 

Lemma 1.4.2. If Go is the real form of the semisimple Lie algebra G, then two above-mentioned forms coincide (up to the nonzero factor). 

One should double all the sides of the parallelepiped II2,..., Iljb+i. As a result, one obtains a new parallelepiped II extended in k directions. Now act with this parallelepiped II upon the point h. As a result, obtain a certain orbit Ti h). It is clear that now this orbit is represented (in action-angle variables on a Liouville torus) by a linear plane which is almost-closed after factorization of the cube to the torus. Farther arguments repeat those of the orientable case. This implies the lemma. 

Lemma 5,2.1 Given a sequencing graph Gm without conditional and loop vertices, the concurrency factor of a subset cf shareable operations Q CV is equal to the clique cover number of the corresponding disjoint compatibility graph Gq. 

The estimates of Lemma 13.5 remain valid in this case. Indeed, it is easy to trace throughout our proof (compare also with the analogous statement in Appendix B of Part I) that an additional -times q [Pg.430]

The key idea for proving Theorem 5.2 is the following lemma, in which we rewrite as P-expectation of a Boltzmaim factor in which the energy... 

The complexity of this statement lies of course in the pre-factor Ce (O tu) and that cannot be avoided, as we explain in in Figure 7.1. The following lemma, from which we will extract the proof of Theorem 7.1, should also be of help for the intuition ... 

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