Abstract
We present an algorithm to calculate generators for the invariant field k(x) G of a linear algebraic group G from the defining equations of G.
This work was motivated by an algorithm of Derksen which allows the computation of the invariant ring of a reductive group using ideal theoretic techniques and the Reynolds operator. The method presented here does not use the Reynolds operator and hence applies to all linear algebraic groups. Like Derksen’s algorithm we start with computing the ideal vanishing on all vectors (ξ,ζ) for which ξ and ζ are on the same orbit. But then we establish a connection of this ideal to the ideal of syzygies the generators of the field k(x) have over the invariant field. From this ideal we can calculate the generators of the invariant field exploiting a field-ideal-correspondence which has been applied to the decomposition of rational mappings before.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Thomas Becker and Volker Weispfenning. Grübner Bases: A Computational Approach to Commutative Algebra. In cooperation with Heinz Kredel. Graduate Texts in Mathematics. Springer, New York, 1993.
David Cox, John Little, and Donal O’Shea. Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer, New York, 1992.
W. Decker and T. De Jong. Gröbner Bases and Invariant Theory. In Bruno Buchberger and Franz Winkler, editors, Gröbner Bases and Applications (Proc. of the Conference 33 Years of Gröbner Bases), volume 251 of London Mathematical Society Lecture Notes Series.Cambridge University Press, 1998
Harm Derksen. Computation of reductive group invariants. Preprint1., 1997
David Eisenbud. Commutative Algebra with a View Toward Algebraic Geomety. Graduate Texts in Mathemathics. Springer, New York, 1995.
P. Gianni, B. Trager, and G. Zacharias. Gröbner bases and primary decomposition of polynomial ideals. Journal of Symbolic Computation, 6:149–167, 1988.
M. Grassl, M. Rötteler, and Th. Beth. Computing Local Invariants of Quantum Bit Systems. Physical Review A, 58(3):1833–1839, September 1998.
Gregor Kemper. An Algorithm to Determine Properties of Field Extensions Lying over a Ground Field. IWR Preprint 93-58, Heidelberg, Oktober 1993.
Gregor Kemper. Das Noethersche Problem und generische Polynome. Dissertation. Universität Heidelberg, August 1994.
Gregor Kemper. Calculating Invariant Rings of Finite Groups over Arbitrary Fields. Journal of Symbolic Computation, 21(3):351–366, März 1996.
Jörn Müller-Quade and Thomas Beth. Computing the Intersection of Finitely Generated Fields. Presented on the ISSAC poster session, 1998. An extended version is submitted for publication.
Jörn Müller-Quade and Martin Rötteler. Deciding Linear Disjointness of Finitely Generated Fields. In Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation. ACM-Press, 1998.
Jörn Müller-Quade, Rainer Steinwandt, and Thomas Beth. An application of Gröbner bases to the decomposition of rational mappings. In Bruno Buchberger and Franz Winkler, editors, Gröbner Bases and Applications (Proc. of the Conference 33 Years of Gröbner Bases), volume 251 of London Mathematical Society Lecture Notes Series. Cambridge University Press, 1998.
L. Smith. Polynomial Invariants of Finite Groups. A. K. Peters, Wellesley, Massachusetts, 1995.
T. A Springer. Aktionen Reduktiver Gruppen auf Varietäten. In H.-P. Kraft, P. Slodowy, and T. A. Springer, editors, Algebraic Transformation Groups and Invariant Theory, pages 3–39. Birkhäuser, 1989.
Bernd Sturmfels. Algorithms in Invariant Theory. Texts and Monographs in Symbolic Computation. Springer, Wien, 1993.
Moss Sweedler. Using Groebner Bases to Determine the Algebraic and Transcendental Nature of Field Extensions: return of the killer tag variables. In Gérard Cohen, Teo Mora, and Oscar Moreno, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes 10th International Symposium, AAECC-10, volume 673 of LNCS, pages 66–75, Berlin; Heidelberg, 1993. Springer.
O. Zariski and P. Samuel. Commutative Algebra, volume 1. Springer, 1958.
O. Zariski and P. Samuel. Commutative Algebra, volume 2. Springer, 1960.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Müller—Quade, J., Beth, T. (1999). Calculating Generators for Invariant Fields of Linear Algebraic Groups. In: Fossorier, M., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1999. Lecture Notes in Computer Science, vol 1719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46796-3_37
Download citation
DOI: https://doi.org/10.1007/3-540-46796-3_37
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66723-0
Online ISBN: 978-3-540-46796-0
eBook Packages: Springer Book Archive