A Constructive Description of SAGBI Bases for Polynomial Invariants of Permutation Groups

Finite SAGBI bases for polynomial invariants of conjugates of alternating groups

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...

Sagbi Bases in Rings of Multiplicative Invariants

Let k be a field and G be a finite subgroup of GLn(Z). We show that the ring of multiplicative invariants k[x±1 1 , . . . , x ±1 n ] G has a finite SAGBI basis if and only if G is generated by reflections.

Sagbi Basis of Permutation Groups and Convex Cones

Bases for structures and permutation groups

Bases, determining sets, metric dimension, . . . The notion of a base, and various combinatorial variants on it, have been rediscovered many times in different parts of combinatorics, especially graph theory: base size has been called fixing number, determining number, rigidity index, etc. Robert Bailey and I have written a survey paper attempting to describe all these and related concepts and ...

Bases for Primitive Permutation Groups and a Conjecture of Babai

A base of a permutation group G is a sequence B of points from the permutation domain such that only the identity of G fixes B pointwise. We show that primitive permutation groups with no alternating composition factors of degree greater than d and no classical composition factors of rank greater than d have a base of size bounded above by a function of d. This confirms a conjecture of Babai. Q...