On Hilbert ideals for a class of $p$-groups in characteristic $p$
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- by Manoj Kummini and Mandira Mondal PDF
- Proc. Amer. Math. Soc. 150 (2022), 145-151 Request permission
Abstract:
Let $p$ be a prime number, $\Bbbk$ a field of characteristic $p$ and $G$ a finite $p$-group. Let $V$ be a finite-dimensional linear representation of $G$ over $\Bbbk$. Write $S = SymV^*$. For a class of $p$-groups which we call generalised Nakajima groups, we prove the following:
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The Hilbert ideal is a complete intersection. As a consequence, for the case of generalised Nakajima groups, we prove a conjecture of Shank and Wehlau (reformulated by Broer) that asserts that if the invariant subring $S^G$ is a direct summand of $S$ as $S^G$-modules then $S^G$ is a polynomial ring.
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The Hilbert ideal has a generating set with elements of degree at most $|G |$. This bound is conjectured by Derksen and Kemper.
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Additional Information
- Manoj Kummini
- Affiliation: Chennai Mathematical Institute, Siruseri, Tamilnadu 603103. India
- MR Author ID: 827227
- ORCID: 0000-0002-4822-0112
- Email: mkummini@cmi.ac.in
- Mandira Mondal
- Affiliation: Chennai Mathematical Institute, Siruseri, Tamilnadu 603103. India
- MR Author ID: 1299300
- ORCID: 0000-0001-6567-3546
- Email: mandiram@cmi.ac.in
- Received by editor(s): November 17, 2020
- Received by editor(s) in revised form: May 21, 2021
- Published electronically: October 25, 2021
- Additional Notes: The authors were supported by the grant CRG/2018/001592 from Science and Engineering Research Board, India and by an Infosys Foundation fellowship.
- Communicated by: Jerzy Weyman
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 145-151
- MSC (2020): Primary 13A50
- DOI: https://doi.org/10.1090/proc/15749
- MathSciNet review: 4335864
Dedicated: In memory of Prof. C. S. Seshadri