Müfit Sezer

Rings of invariants for modular representations of the Klein four group

We study the rings of invariants for the indecomposable modular representations of the Klein four group. For each such representation we compute the Noether number and give minimal generating sets for the Hilbert ideal and the field of fractions. We observe that, with the exception of the regular representation, the Hilbert ideal for each of these representations is a complete intersection.

SEPARATING INVARIANTS FOR THE KLEIN FOUR GROUP AND CYCLIC GROUPS

We consider indecomposable representations of the Klein four group over a field of characteristic 2 and of a cyclic group of order pm with p, m coprime over a field of characteristic p. For each representation, we explicitly describe a separating set in the corresponding ring of invariants. Our construction is recursive and the separating sets we obtain consist of almost entirely orbit sums and products.

Invariants of the dihedral group D2p in characteristic two

We consider finite dimensional representations of the dihedral group D 2 p over an algebraically closed field of characteristic two where p is an odd prime and study the degrees of generating and separating polynomials in the corresponding ring of invariants. We give an upper bound for the degrees of the polynomials in a minimal generating set that does not depend on p when the dimension of the representation is sufficiently large. We also show that p + 1 is the minimal number such that the invariants up to that degree always form a separating set. We also give an explicit description of a separating set.

Gröbner bases for the Hilbert ideal and coinvariants of the dihedral group D2p

obner bases MSC (2010) 13A50 We consider a finite dimensional representation of the dihedral group D2 p over a field of characteristic two where p is an odd integer and study the corresponding Hilbert ideal IH .W e show thatIH has a universal Gr¨ obner basis consisting of invariants and monomials only. We provide sharp bounds for the degree of an element in this basis and in a minimal generating set for IH . We also compute the top degree of coinvariants when p is prime.

Explicit separating invariants for cyclic P-groups

We consider a finite-dimensional indecomposable modular representation of a cyclic p-group and we give a recursive description of an associated separating set: We show that a separating set for a representation can be obtained by adding, to a separating set for any subrepresentation, some explicitly defined invariant polynomials. Meanwhile, an explicit generating set for the invariant ring is known only in a handful of cases for these representations.

On The Top Degree of Coinvariants

For a finite group

The Noether Map I

G

The noether map II

acting faithfully on a finite dimensional

On Cohen–Macaulayness and depth of ideals in invariant rings

F

Gröbnerian Dickson polynomials

-vector space