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By Neal W. Stoltzfus
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Extra resources for Unraveling the integral knot concordance group
Example text
JC„] is completely determined by its values on the variables Xi. By the above formulas, g o ^ o g~'^ (xj) = bij = J^k ^i^jrS'^J^'' ^^^^^ n the statement. The above formulas prove that the space of derivatives behaves as the dual of the space of the variables and that the action of the group is by inverse transpose. This of course has an intrinsic meaning: if V is a vector space and V{V) the ring of polynomials on V, we have that V* c V{V) are the linear polynomials. The space V can be identified intrinsically with the space spanned by the derivatives.
Mn-x)' Thus if the degree is less than n, the map 7r„ maps these basis elements into distinct basis elements. 3. ^-iy-'eiifn-i. oriA, = 0 = Y^(-iyeiifn-i, i=\ i=0 By the previous lemma this identity also remains valid for symmetric functions in more than n variables and gives the required recursion. 2 Symmetric Polynomials It is actually a general fact that symmetric functions can be expressed as polynomials in the elementary synmietric functions. We will now discuss an algorithmic proof. , jCjt.
Thus we have the corresponding two actions on F[G] by (h, k)f(g) = f(h~^gk) and we may view the right action as symmetries of the left action and conversely. Sometimes it is convenient to write ^f^ = (h,k)f to stress the left and right actions. After these basic examples we give a general definition: Definition 1. Given a vector space V over a field F (or more generally a module), we say that an action of a group G on V is linear if every element of G induces a linear transformation on V. A linear action of a group is also called a linear representation;^ a vector space V that has a G-action is called a G-module.