20,209 research outputs found
Separating algebras and finite reflection groups
A separating algebra is, roughly speaking, a subalgebra of the ring of
invariants whose elements distinguish between any two orbits that can be
distinguished using invariants. In this paper, we introduce a geometric notion
of separating algebra. This allows us to prove that only groups generated by
reflections may have polynomial separating algebras, and only groups generated
by bireflections may have complete intersection separating algebras.Comment: 12 pages, corrected yet another typ
Generic separating sets for 3D elasticity tensors
We define what is a generic separating set of invariant functions (a.k.a. a
weak functional basis) for tensors. We produce then two generic separating sets
of polynomial invariants for 3D elasticity tensors, one made of 19 polynomials
and one made of 21 polynomials (but easier to compute) and a generic separating
set of 18 rational invariants. As a byproduct, a new integrity basis for the
fourth-order harmonic tensor is provided
Minimal generating and separating sets for O(3)-invariants of several matrices
Given an algebra of polynomial invariants of an action of the group
over the vector space , a subset of is called separating if
separates all orbits that can be separated by . A minimal
separating set is found for some algebras of matrix invariants of several
matrices over an infinite field of arbitrary characteristic different from two
in case of the orthogonal group. Namely, we consider the following cases:
1) -invariants of two matrices;
2) -invariants of skew-symmetric matrices;
3) -invariants of two skew-symmetric matrices;
4) -invariants of two symmetric matrices.
A minimal generating set is also given for the algebra of orthogonal
invariants of three symmetric matrices.Comment: 11 page
Degree bounds for separating invariants
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) in case H is normal.
Moreover, we calculate b(G) for some specific finite groups.Comment: 11 page
Invariants of the dihedral group in characteristic two
We consider finite dimensional representations of the dihedral group
over an algebraically closed field of characteristic two where is an odd
integer 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 when the
dimension of the representation is sufficiently large. We also show that
is the minimal number such that the invariants up to that degree always form a
separating set. As well, we give an explicit description of a separating set
when is prime.Comment: 7 page
Degree bound for separating invariants of abelian groups
It is proved that the universal degree bound for separating polynomial
invariants of a finite abelian group (in non-modular characteristic) is
strictly smaller than the universal degree bound for generators of polynomial
invariants, unless the goup is cyclic or is the direct product of even
order cyclic groups where the number of two-element direct factors is not less
than the integer part of the half of . A characterization of separating sets
of monomials is given in terms of zero-sum sequences over abelian groups
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