If V is a representation of a linear algebraic group G, a set S of
G-invariant regular functions on V is called separating if the following holds:
If two elements v,v' from V can be separated by an invariant function, then
there is an f from S such that f(v) is different from f(v'). It is known that
there always exist finite separating sets. Moreover, if the group G is finite,
then the invariant functions of degree <= |G| form a separating set. We show
that for a non-finite linear algebraic group G such an upper bound for the
degrees of a separating set does not exist. If G is finite, we define b(G) to
be the minimal number d such that for every G-module V there is a separating
set of degree less or equal to d. We show that for a subgroup H of G we have
b(H) <= b(G) <= [G:H] b(H)$, and that b(G) <= b(G/H) b(H)$ in case H is normal.
Moreover, we calculate b(G) for some specific finite groups.Comment: 11 page
Abstract. We fix a field k of characteristic p. For a finite group G denote by δ(G) and σ(G) respectively the minimal number d, such that for any finite dimensional representation V of G over k and any v ∈ V G \ {0} or v ∈ V \ {0} respectively, there exists a homogeneous invariant f ∈ k[V ] G of positive degree at most d such that f (v) = 0. Let P be a Sylow-p-subgroup of G (which we take to be trivial if the group order is not divisble by p). We show that δ(G) = |P |., where l is the smallest prime divisor of |G|. These results extend known results in the non-modular case to the modular case.
Abstract. In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.
We explicitly construct a finite set of separating invariants for the basic Ga-actions. These are the finite dimensional indecomposable rational linear representations of the additive group Ga of a field of characteristic zero, and their invariants are the kernel of the Weitzenböck derivation Dn = x 0 ∂ ∂x 1 + . . . + x n−1 ∂ ∂xn .
In Germany there are 16 million people with a migration background, one in five of the total population. There are relatively few migrant mortality studies in Germany, which is primarily due to the restricted quantity and quality of existing data. The official migrant death statistics for Germany suffer from incomplete migrant population stock data due to non-registered remigration events. After the German census in 2011 especially the migrant stock data was adjusted downwards, and therefore realistic estimates of the migrant mortality risk and the healthy-migrant-effect are possible. Between 2010 and 2013 mortality risks of foreigners rose strongly due to the census corrections of the migrant population. However, the risks for adults and pensioners still lie below the risks for Germans in the same age groups. The lower risks indicate a healthy-migrant-effect, whicht was primarily effective shortly after the immigration event. Analysis based on data from the Statutory Pension Insurance (GRV) shows higher migrant mortality risks in the age group from 65 to 84. In that age group there are supposedly a lot of people, who immigrated to Germany in the context of the guest worker recruitment in the 1950s to 1970s and who had hard working conditions in their lifetimes. Their mortality risk, therefore, increased in the long-term perspective. In the future the lack of data in the migrant population will again rise due to unregistered remigration. Alternative databases need to be used for migrant mortality analyses.
Let G be a linear algebraic group over an algebraically closed field k acting rationally on a G-module V with N G,V its null-cone. Let δ(G, V ) and σ(G, V ) denote the minimal number d such that for every v ∈ V G \ N G,V and v ∈ V \ N G,V , respectively, there exists a homogeneous invariant f of positive degree at most d such that f (v) = 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V . For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL 2 (k) that contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G 0 is unipotent. Our results also lead to a more elementary proof that βsep(G) is finite if and only if G is finite.
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