SAGBI bases for rings of invariant Laurent polynomials
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- by Alexander Duncan and Zinovy Reichstein PDF
- Proc. Amer. Math. Soc. 137 (2009), 835-844 Request permission
Abstract:
Let $k$ be a field, let $L_n = k[x_1^{\pm 1}, \dots , x_n^{\pm 1}]$ be the Laurent polynomial ring in $n$ variables and let $G$ be a finite group of $k$-algebra automorphisms of $L_n$. We give a necessary and sufficient condition for the ring of invariants $L_n^G$ to have a SAGBI basis. We show that if this condition is satisfied, then $L_n^G$ has a SAGBI basis relative to any choice of coordinates in $L_n$ and any term order.References
- Winfried Bruns and Joseph Gubeladze, Polytopes, rings and k-theory, http://math.sfsu.edu/gubeladze/publications/kripo/kripo.pdf.
- David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. An introduction to computational algebraic geometry and commutative algebra. MR 1189133, DOI 10.1007/978-1-4757-2181-2
- Leonard Eugene Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with $n$ Distinct Prime Factors, Amer. J. Math. 35 (1913), no. 4, 413–422. MR 1506194, DOI 10.2307/2370405
- Manfred Göbel, The optimal lower bound for generators of invariant rings without finite SAGBI bases with respect to any admissible order, Discrete Math. Theor. Comput. Sci. 3 (1999), no. 2, 65–70. MR 1695195
- Manfred Göbel, Finite SAGBI bases for polynomial invariants of conjugates of alternating groups, Math. Comp. 71 (2002), no. 238, 761–765. MR 1885626, DOI 10.1090/S0025-5718-01-01405-3
- Shigeru Kuroda, The infiniteness of the SAGBI bases for certain invariant rings, Osaka J. Math. 39 (2002), no. 3, 665–680. MR 1932287
- Martin Lorenz, Multiplicative invariant theory, Encyclopaedia of Mathematical Sciences, vol. 135, Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, VI. MR 2131760
- Zinovy Reichstein, SAGBI bases in rings of multiplicative invariants, Comment. Math. Helv. 78 (2003), no. 1, 185–202. MR 1966757, DOI 10.1007/s000140300008
- Lorenzo Robbiano and Moss Sweedler, Subalgebra bases, Commutative algebra (Salvador, 1988) Lecture Notes in Math., vol. 1430, Springer, Berlin, 1990, pp. 61–87. MR 1068324, DOI 10.1007/BFb0085537
- Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949, DOI 10.1090/ulect/008
- Mohammed Tesemma, On multiplicative invariants of finite reflection groups, Comm. Algebra 35 (2007), no. 7, 2258–2274. MR 2331844, DOI 10.1080/00927870701302255
- N. M. Thiéry and S. Thomassé, Convex cones and SAGBI bases of permutation invariants, Invariant theory in all characteristics, CRM Proc. Lecture Notes, vol. 35, Amer. Math. Soc., Providence, RI, 2004, pp. 259–263. MR 2066473, DOI 10.1090/crmp/035/19
Additional Information
- Alexander Duncan
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
- MR Author ID: 854896
- Email: duncan@math.ubc.ca
- Zinovy Reichstein
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
- MR Author ID: 268803
- Email: reichst@math.ubc.ca
- Received by editor(s): February 6, 2008
- Received by editor(s) in revised form: February 28, 2008
- Published electronically: September 15, 2008
- Additional Notes: The first author was partially supported by an NSERC Canada Graduate Scholarship.
The second author was partially supported by NSERC Discovery and Accelerator Supplement grants - Communicated by: Martin Lorenz
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 835-844
- MSC (2000): Primary 13A50, 13P99
- DOI: https://doi.org/10.1090/S0002-9939-08-09538-5
- MathSciNet review: 2457421