Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T23:22:19.450Z Has data issue: false hasContentIssue false
  • English
  • Français

Degrees of Regular Sequences With a Symmetric Group Action

Part of: General commutative ring theory Representation theory of groups

Published online by Cambridge University Press:  07 January 2019

Federico Galetto ,
Anthony Vito Geramita  and
David Louis Wehlau
Federico Galetto
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton ON L8S 4K1 Email: [email protected]
Anthony Vito Geramita
Affiliation:
Department of Mathematics, Queen’s University, Kingston ON K7L 3N6 Email: [email protected]
David Louis Wehlau
Affiliation:
Department of Mathematics and Computer Science, Royal Military College, Kingston ON K7K 7B4 Email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees.

Keywords

complete intersectionsymmetric groupregular sequence

MSC classification

Primary: 13A02: Graded rings
Secondary: 13A50: Actions of groups on commutative rings; invariant theory 20C30: Representations of finite symmetric groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 3 , June 2019 , pp. 557 - 578
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The authors gratefully acknowledge the partial support of NSERC for this work.

References

Blokhuis, Aart, Brouwer, Andries E., and Szőnyi, Tamás, Proof of a conjecture by Ðoković on the Poincaré series of the invariants of a binary form . Indag. Math. (N.S.) 24(2013), no. 4, 766–773. https://doi.org/10.1016/j.indag.2012.12.004.Google Scholar
Broué, Michel, Introduction to complex reflection groups and their braid groups . Lecture Notes in Mathematics, 1988. Springer-Verlag, Berlin, 2010. https://doi.org/10.1007/978-3-642-11175-4.Google Scholar
Bruns, Winfried and Herzog, Jügen, Cohen-Macaulay rings . Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993.Google Scholar
Chen, Ri-Xiang, On two classes of regular sequences . J. Commut. Algebra 8(2016), no. 1, 29–42. https://doi.org/10.1216/JCA-2016-8-1-29.Google Scholar
Conca, Aldo, Krattenthaler, Christian, and Watanabe, Junzo, Regular sequences of symmetric polynomials . Rend. Semin. Mat. Univ. Padova 121(2009), 179–199. https://doi.org/10.4171/RSMUP/121-11.Google Scholar
Dixmier, Jacques, Quelques résultats et conjectures concernant les séries de Poincaré des invariants des formes binaires . In: Séminaire d’algèbre Paul Dubreil et Marie-Paule Malliavin, 36ème année (Paris, 1983–1984). Lecture Notes in Mathematics, 1146. Springer, Berlin, 1985, pp. 127–160. https://doi.org/10.1007/BFb0074537.Google Scholar
Ðoković, Dragomir Ž., A heuristic algorithm for computing the Poincaré series of the invariants of binary forms . Int. J. Contemp. Math. Sci. 1(2006), 557–568. https://doi.org/10.12988/ijcms.2006.06059.Google Scholar
Eisenbud, David, Commutative algebra . Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. https://doi.org/10.1007/978-1-4612-5350-1.Google Scholar
Fulton, William, Young tableaux . London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 1997.Google Scholar
Galetto, Federico, Geramita, Anthony V., and Wehlau, David L., Symmetric complete intersections . Comm. Algebra 46(2018), 2194–2204. https://doi.org/10.1080/00927872.2017.1372453.Google Scholar
Goodman, Roe and Wallach, Nolan R., Symmetry, representations, and invariants . Graduate Texts in Mathematics, 255. Springer, Dordrecht, 2009. https://doi.org/10.1007/978-0-387-79852-3.Google Scholar
Granville, Andrew, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers . In: Organic mathematics. CMS Conf. Proc., 20. Amer. Math. Soc., Providence, RI, 1997, pp. 253–276.Google Scholar
Grayson, Daniel R. and Stillman, Michael E., Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/.Google Scholar
Kumar, Neeraj and Martino, Ivan, Regular sequences of power sums and complete symmetric polynomials . Matematiche (Catania), 67(2012), no. 1, 103–117.Google Scholar
Lam, Tsit Yuen and Leung, Ka Hin, On vanishing sums of roots of unity . J. Algebra 224(2000), no. 1, 91–109. https://doi.org/10.1006/jabr.1999.8089.Google Scholar
Lang, Serge, Algebra . Third edition. Graduate Texts in Mathematics, 211. Springer-Verlag, New York, 2002. https://doi.org/10.1007/978-1-4613-0041-0.Google Scholar
Littelmann, Peter and Procesi, Claudio, On the Poincaré series of the invariants of binary forms . J. Algebra 133(1990), no. 2, 490–499. https://doi.org/10.1016/0021-8693(90)90284-U.Google Scholar
Alfonsín, Jorge Luis Ramírez, The Diophantine Frobenius problem . Oxford Lecture Series in Mathematics and its Applications, 30. Oxford University Press, Oxford, 2005. https://doi.org/10.1093/acprof:oso/9780198568209.001.0001.Google Scholar
Sagan, Bruce E., The symmetric group . Second edition. Graduate Texts in Mathematics, 203. Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4757-6804-6.Google Scholar
Stanley, Richard P., Enumerative combinatorics . Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999. https://doi.org/10.1017/CBO9780511609589.Google Scholar