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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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On the exotic Grassmannian and its nilpotent variety
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by Lucas Fresse and Kyo Nishiyama PDF
Represent. Theory 20 (2016), 451-481 Request permission

Abstract:

Given a decomposition of a vector space $V=V_1\oplus V_2$, the direct product $\mathfrak {X}$ of the projective space $\mathbb {P}(V_1)$ with a Grassmann variety $\mathrm {Gr}_k(V)$ can be viewed as a double flag variety for the symmetric pair $(G,K)=(\mathrm {GL}(V),\mathrm {GL}(V_1)\times \mathrm {GL}(V_2))$. Relying on the conormal variety for the action of $K$ on $\mathfrak {X}$, we show a geometric correspondence between the $K$-orbits of $\mathfrak {X}$ and the $K$-orbits of some appropriate exotic nilpotent cone. We also give a combinatorial interpretation of this correspondence in some special cases. Our construction is inspired by a classical result of Steinberg (1976) and by the recent work of Henderson and Trapa (2012) for the symmetric pair $(\mathrm {GL}(V),\mathrm {Sp}(V))$.
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Additional Information
  • Lucas Fresse
  • Affiliation: Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France
  • MR Author ID: 875745
  • Email: lucas.fresse@univ-lorraine.fr
  • Kyo Nishiyama
  • Affiliation: Department of Physics and Mathematics, Aoyama Gakuin University, Fuchinobe 5-10-1, Sagamihara 252-5258, Japan
  • MR Author ID: 207972
  • Email: kyo@gem.aoyama.ac.jp
  • Received by editor(s): April 6, 2016
  • Received by editor(s) in revised form: October 9, 2016
  • Published electronically: November 28, 2016
  • Additional Notes: The first author was supported by the ISF Grant Nr. 797/14 and by the ANR project NilpOrbRT (ANR-12-PDOC-0031).
    The second author was supported by JSPS KAKENHI Grant Numbers #25610008 and #16K05070.
  • © Copyright 2016 American Mathematical Society
  • Journal: Represent. Theory 20 (2016), 451-481
  • MSC (2010): Primary 14L30; Secondary 14L35, 14M15, 17B08
  • DOI: https://doi.org/10.1090/ert/489
  • MathSciNet review: 3576071