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A geometric interpretation of the characteristic polynomial of reflection arrangements


Authors: Mathias Drton and Caroline J. Klivans
Journal: Proc. Amer. Math. Soc. 138 (2010), 2873-2887
MSC (2010): Primary 51F15, 05E15, 20F55, 62H15
DOI: https://doi.org/10.1090/S0002-9939-10-10369-4
Published electronically: April 8, 2010
MathSciNet review: 2644900
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Abstract: We consider projections of points onto fundamental chambers of finite real reflection groups. Our main result shows that for groups of types $ A_n$, $ B_n$, and $ D_n$, the coefficients of the characteristic polynomial of the reflection arrangement are proportional to the spherical volumes of the sets of points that are projected onto faces of a given dimension. We also provide strong evidence that the same connection holds for the exceptional, and thus all, reflection groups. These results naturally extend those of De Concini and Procesi, Stembridge, and Denham, which establish the relationship for 0-dimensional projections. This work is also of interest to the field of order-restricted statistical inference, where projections of random points play an important role.


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Additional Information

Mathias Drton
Affiliation: Department of Statistics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637-1514

Caroline J. Klivans
Affiliation: Departments of Mathematics and Computer Science, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637-1538

DOI: https://doi.org/10.1090/S0002-9939-10-10369-4
Keywords: Characteristic polynomial, Coxeter group, hyperplane arrangement, order-restricted statistical inference, reflection group
Received by editor(s): June 11, 2009
Published electronically: April 8, 2010
Additional Notes: The first author was partially supported by NSF grant DMS-0746265 and an Alfred P. Sloan Research Fellowship.
Communicated by: Jim Haglund
Article copyright: © Copyright 2010 American Mathematical Society

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