American Mathematical Society's A crash course on Kleinian groups; lectures given at a PDF

# American Mathematical Society's A crash course on Kleinian groups; lectures given at a PDF

By American Mathematical Society

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The article of this ebook is the quantum mechanism that enables the macroscopic quantum coherence of a superconducting condensate to withstand to the assaults of extreme temperature. approach to this basic challenge of recent physics is required for the layout of room temperature superconductors, for controlling the decoherence results within the quantum desktops and for the certainty of a potential function of quantum coherence in dwelling subject that's debated this day in quantum biophysics.

Extra info for A crash course on Kleinian groups; lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco

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7 Commuting Actions We need another generality. , assume that we have two groups G and H acting on the same set X. We say that the two actions commute if gh{x) = hg{x) for allx e X, g e G and he H. This means that every element of G gives rise to an H equivariant map (or we can reverse the roles of G and / / ) . It also means that we really have an action of the product group G x H onX given by (g, h)x = ghx. In this case, we easily see that if a function / is G-invariant and h e H, then hf is also G-invariant.

We have seen that: Proposition 2. The space A of antisymmetric polynomials is a free rank 1 module over the ring S of symmetric polynomials generated by V{x) or A = V(x)S. In particular, any integral basis of A gives, dividing by V(x), an integral basis of S. In this way we will presently obtain the Schur functions. To understand the construction, let us make a fairly general discussion. , JC„], let US consider the basis given by the monomials (which are permuted by 5„). Recall that the orbits of monomials are indexed by non-increasing sequences of nonnegative integers.

In general, a direct sum of vector spaces U = ^f^QUk is called a graded vector space. A subspace W of L^ is called homogeneous, if, setting Wj := WH Ui, we have The space of polynomials is thus a graded vector space P[V] = 0 ^ o ^ ^ [ ^ ] One has immediately {gf){otv) ~ f{ag~^v) = a^{gf){v), which has an important consequence: Theorem. If a polynomial f is an invariant (under some linear group action), then its homogeneous components are also invariant. Proof. Let f = Yl fi ^^ ^^^ decomposition of / into homogeneous components, gf = J2 Sfi is the decomposition into homogeneous components of gf If / is invariant / = gf then // = gfi for each / since the decomposition into homogeneous D components is unique.