Some Conjectures of Mine and Others
These conjectures are all original but I make
no claims of priority.
The conjectures concern modular invariant theory and many of them
the Noether number. They are expressed using standard notation
False Conjecture 1: Let G be a finite group.
Let R denote k[V]G
and let I denote the image of the transfer homomorphism. I had conjectured
that b(R/I) <=
|G| but Peter Fleischmann, Gregor Kemper and Jim Shank have just recently
produced a counter example to this.
Conjecture 2: Let G be a finite p-group where k is of
characteristic p > 0.
If Im TrG is a principal ideal
then k[V]G is a polynomial ring.
Jim Shank and I proved the
converse of this conjecture when k is the prime field
Bram Broer has recently proved the converse of the conjecture for any
field k of characteristic p.
No 3's Conjecture: Let Vp denote the regular
representation of the cyclic group Z/p over the finite field of
prime order Fp. Jim Shank and I had conjectured
b(Fp[Vp]Z/p) = 2p-3.
We had proven that b(Fp[Vp]Z/p)
In a recent preprint (Fall 2005) P. Fleischmann, M. Sezer, R.J. Shank and C.F. Woodcock
prove that b(Fp[Vp]Z/p)
is indeed 2p-3.
I conjecture that
is generated by Norms together with the orbit sums of the monomials (with respect to the usual
permutation basis) which have no
exponent exceeding 2.
Conjecture 4 If V is a representation of G and W is a subrepresentation of V then
b(k[W]G) <= b(k[V]G).
Jim Shank and I have proven this conjecture for G=Z/p.
Conjecture 5 Let V be a representation of a p-group with a polynomial ring of invariants.
I conjecture that it is always possible to take a (non-lnear) norm of a linear form as one of
the minimal generators for the ring of invariants. I even think this is true without the
assumption that the ring of invariants is polynomial. This more general form is true for cyclic
proofs (or disproofs) are welcome!
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