By Louis Comtet
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Extra info for Analyse Combinatoire
Our input will be a black box which evaluates the Reynolds operator “ ” of , and our output will be a set of primary and secondary invariants as in Sect. 3. The reason for this assumption is that the knowledge of the full group might not be necessary to compute the invariants: it suffices to know the Reynolds operator. We begin with a description of six commutative algebra subroutines based on Buchberger’s method. The first four of these algorithms are well known, and they are discussed in practically every introduction to Gröbner bases theory.
5), there exists a finite set ff1 ; f2 ; : : : ; fm g CŒx of homogeneous invariants which generates the ideal I . , no smaller set of homogeneous invariants generates I . We need to prove that m D n, or, equivalently, that the invariants f1 ; f2 ; : : : ; fm are algebraically independent over C. Our proof is by contradiction. f1 ; f2 ; : : : ; fm / D 0 in CŒx. f1 ; f2 ; : : : ; fm / have the same degree d WD i1 C i2 C : : : C in . f1 ; f2 ; : : : ; fm / 2 CŒx : @yi Each gi is either 0 or of degree d deg fi .
I/ . Let r be the number of reflections in and therefore in H . 5 we have jj D d1 d2 : : : dn D e1 e2 : : : en D jH j, and hence H D . G The “only-if” part is useful in that it proves that most invariant rings are not polynomial rings. 6 (Twisted symmetric polynomials). C n /, and consider its representation WD fsign. / W 2 Sn g. We call the elements of the invariant ring CŒx twisted symmetric polynomials. Note that a homogeneous polynomial f is twisted symmetric if and only if f B D sign.
Analyse Combinatoire by Louis Comtet