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Degrees of Regular Sequences With a Symmetric Group Action

Published online by Cambridge University Press:  07 January 2019

Federico Galetto
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton ON L8S 4K1 Email: galettof@math.mcmaster.ca
Anthony Vito Geramita
Affiliation:
Department of Mathematics, Queen’s University, Kingston ON K7L 3N6 Email: tony@mast.queensu.ca
David Louis Wehlau
Affiliation:
Department of Mathematics and Computer Science, Royal Military College, Kingston ON K7K 7B4 Email: wehlau@rmc.ca
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Abstract

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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.

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Article
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, 766773. https://doi.org/10.1016/j.indag.2012.12.004.CrossRefGoogle 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, 2942. https://doi.org/10.1216/JCA-2016-8-1-29.CrossRefGoogle Scholar
Conca, Aldo, Krattenthaler, Christian, and Watanabe, Junzo, Regular sequences of symmetric polynomials . Rend. Semin. Mat. Univ. Padova 121(2009), 179199. https://doi.org/10.4171/RSMUP/121-11.CrossRefGoogle 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. 127160. 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), 557568. https://doi.org/10.12988/ijcms.2006.06059.CrossRefGoogle 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), 21942204. https://doi.org/10.1080/00927872.2017.1372453.CrossRefGoogle 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. 253276.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, 103117.Google Scholar
Lam, Tsit Yuen and Leung, Ka Hin, On vanishing sums of roots of unity . J. Algebra 224(2000), no. 1, 91109. https://doi.org/10.1006/jabr.1999.8089.CrossRefGoogle 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, 490499. https://doi.org/10.1016/0021-8693(90)90284-U.CrossRefGoogle 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
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