1998
DOI: 10.1006/jsco.1998.0210
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A Constructive Description of SAGBI Bases for Polynomial Invariants of Permutation Groups

Abstract: Let R be a commutative ring with 1, let R[X 1 , . . . , Xn] be the polynomial ring in X 1 , . . . , Xn over R, and let G be an arbitrary group of permutations of {X 1 , . . . , Xn}. This note presents a detailed analysis and a constructive combinatorial description of SAGBI bases for the R-algebra of G-invariant polynomials. Our main result is a ground ring independent characterization of all rings of polynomial invariants of permutation groups G having a finite SAGBI basis. Definition 3.1. Let B be a SAGBI (S… Show more

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Cited by 34 publications
(8 citation statements)
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“…Unfortunately, even a finitely generated subalgebra does not necessarily have a finite SAGBI basis. In fact even the ring of invariants of a finite group may fail to have a SAGBI basis (see Göbel, 1995, Lemma 2.1;Göbel, 1998or Sturmfels, 1996. The characterization of subalgebras which admit a finite SAGBI basis is an important open problem.…”
Section: Introductionmentioning
confidence: 97%
“…Unfortunately, even a finitely generated subalgebra does not necessarily have a finite SAGBI basis. In fact even the ring of invariants of a finite group may fail to have a SAGBI basis (see Göbel, 1995, Lemma 2.1;Göbel, 1998or Sturmfels, 1996. The characterization of subalgebras which admit a finite SAGBI basis is an important open problem.…”
Section: Introductionmentioning
confidence: 97%
“…, x n ]. For example, in [3,Lemma 5.6] he showed that the ring of invariants k[x] G has a finite SAGBI basis under the usual lexicographic order if and only if G ∼ = S n 1 × · · · × S n k for some partition n = n 1 + · · · + n k . He further conjectured in [4] that the same should be true for an arbitrary monomial order.…”
Section: Introductionmentioning
confidence: 99%
“…SAGBI bases were introduced independently by Robbiano and Sweedler [12] and Kapur and Madlener [10]. In general the ring of invariants of a finite group may fail to have a finite SAGBI basis (see Göbel [7, Lemma 2.1], Göbel [8] or Sturmfels [18,Example 11.2]). However, it was shown by Shank and Wehlau in [16,Corollary 3.3] that the ring of invariants of a p-group in characteristic p always has a finite SAGBI basis.…”
Section: Introductionmentioning
confidence: 99%