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Abstract We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalised exponents. Our refinement gives a uniform proof and generalisation of a recent theorem of the second author.
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The Jacobian, Reflection Arrangement and Discriminant for Reflection Hopf Algebras
Abstract We study finite-dimensional semisimple Hopf algebra actions on noetherian connected graded Artin–Schelter regular algebras and introduce definitions of the Jacobian, the reflection arrangement, and the discriminant in a noncommutative setting.
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- PAR ID:
- 10321585
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2021
- Issue:
- 13
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We study normal reflection subgroups of complex reflection groups. Our point of view leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalized exponents. Our refinement gives a uniform proof and generalization of a recent theorem of the second author.more » « less
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Tammets, K; Sosnovsky, S; Ferreira_Mello, R; Pishtari, G; Nazaretsky, T (Ed.)The increasing use of AI systems opens up opportunities to use learner and teacher-related data for reflection, supporting instruction and professional growth. However, teacher-related data is rarely collected in technology-enhanced learning environments. Understanding teacher preferences can inform the design of AI reflection systems that align with their priorities. The current survey study explores how K-12 teachers' reflection practices and preferences for analytic data types (e.g., learning-related vs. engagement-related) vary by experience, an aspect often overlooked in prior works. Findings from our survey study with N=100 teachers in the U.S. revealed significant differences. Less experienced teachers focus on classroom behavior and prefer analytics on classroom issues, while experienced teachers prefer broader strategies to improve student learning and favor sustained data analysis. Our results reflect the need to consider teachers’ experience level by tuning the complexity and concreteness of analytic recommendations to support ongoing professionalization.more » « less
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnz380
