Let Cp denote the cyclic group of order p where p ≥ 3 is prime. We denote by Vn the indecomposable n dimensional representation of Cp over a field F of characteristic p. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants F[V2 ⊕ V2 ⊕ V3]p .

It is well-known, that the ring C[X1, . . . , Xn]n of polynomial invariants of the alternating group An has no finite SAGBI basis with respect to the lexicographical order for any number of variables n ≥ 3. This note proves the existence of a nonsingular matrix δn ∈ GL(n,C) such that the ring of polynomial invariants C[X1, . . . ,Xn] δn n , where An n denotes the conjugate of An with respect to...

)( was investigated. It turned out that only invariant rings of direct products of symmetric groups have a finite SAGBI basis, which is then, in addition, multilinear. Of course, it would be of interest to have such a strong characterization with respect to any other admissible order [4, 6]. To achieve this seems to be all but trivial. One step towards the understanding of the behavior of SAGBI...

Let k be a field, Ln = k[x ±1 1 , . . . , x ±1 n ] be the Laurent polynomial ring in n variables and G be a group of k-algebra automorphisms of Ln. We give a necessary and sufficient condition for the ring of invariants Ln to have a SAGBI basis. We show that if this condition is satisfied then Ln has a SAGBI basis relative to any choice of coordinates in Ln and any term order.

The maximal minors of a p× (m+p)-matrix of univariate polynomials of degree n with indeterminate coefficients are themselves polynomials of degree np. The subalgebra generated by their coefficients is the coordinate ring of the quantum Grassmannian, a singular compactification of the space of rational curves of degree np in the Grassmannian of p-planes in (m + p)-space. These subalgebra generat...

We introduce canonical bases for subalgebras of quotients of the commutative and non-commutative polynomial ring. A more complete exposition can be found in 4]. Canonical bases for subalgebras of the commutative polynomial ring were introduced by Kapur and Madlener (see 2]), and independently by Robbiano and Sweedler ((5]). Some notes on the non-commutative case can be found in 3]. Using the la...

Our interest in the subject of this paper is inspired by Hong (1998), where Hoon Hong addresses the problem of the behavior of Gröbner bases under composition of polynomials. More precisely, let Θ be a set of polynomials, as many as the variables in our polynomial ring. The question then is under which conditions on these polynomials it is true that for an arbitrary Gröbner basis G (with respec...

The classical reduction techniques of bifurcation theory, Liapunov–Schmidt reduction and centre manifold reduction, are investigated where symmetry is present. The symmetry is given by the action of a finite or continuous group. The symmetry is exploited systematically by using the algebraic structure of the module of equivariant polynomial tuples. We generalize the concept of SAGBI-bases to mo...

In this paper we examine subalgebras on two generators in the univariate polynomial ring. A set, S, of polynomials in a subalgebra of a polynomial ring is called a canonical basis (also referred to as SAGBI basis) for the subalgebra if all lead monomials in the subalgebra are products of lead monomials of polynomials in S. In this paper we prove that a pair of polynomials {f, g} is a canonical ...