Finite SAGBI bases for polynomial invariants of conjugates of alternating groups. (English) Zbl 0994.13003
Summary: It is well-known, that the ring \(\mathbb{C} [X_1,\dotsc,X_n]^{A_n}\) of polynomial invariants of the alternating group \(A_n\) has no finite SAGBI basis with respect to the lexicographical order for any number of variables \(n \geq 3\). This note proves the existence of a non-singular matrix \(\delta_n \in GL(n,\mathbb{C})\) such that the ring of polynomial invariants \(\mathbb{C} [X_1,\dotsc,X_n]^{A_n^{\delta_n}}\), where \(A_n^{\delta_n}\) denotes the conjugate of \(A_n\) with respect to \(\delta_n\), has a finite SAGBI basis for any \(n \geq 3\).
MSC:
| 13A50 | Actions of groups on commutative rings; invariant theory |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 12Y05 | Computational aspects of field theory and polynomials (MSC2010) |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
Keywords:
algorithmic invariant theory; finite SAGBI bases; rewriting techniques; polynomial invariants of the alternating groupReferences:
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