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

By American Mathematical Society

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**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**

**Sample text**

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.