Abstract
We construct the finite dimensional simple integral modules for the (degenerate) affine Hecke–Clifford algebra (AHCA), \({\mathcal{H}_{\mathcal{C}\ell}^{{\rm aff}}(d)}\). Our construction includes an analogue of Zelevinsky’s segment representations, a complete combinatorial description of the simple calibrated \({\mathcal{H}_{\mathcal{C}\ell}^{{\rm aff}}(d)}\)-modules, and a classification of the simple integral \({\mathcal{H}_{\mathcal{C}\ell}^{{\rm aff}}(d)}\)-modules. Our main tool is an analogue of the Arakawa–Suzuki functor for the Lie superalgebra \({\mathfrak{q}(n)}\).
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Acknowledgments
The work presented in this paper was begun while the second author visited the Mathematical Sciences Research Institute in Berkeley, CA. He would like to thank the administration and staff of MSRI for their hospitality and especially the organizers of the “Combinatorial Representation Theory” and “Representation Theory of Finite Groups and Related Topics” programs for providing an exceptionally stimulating semester. We would like to thank Mikhail Khovanov for suggesting we consider a super analogue of the Arakawa–Suzuki functor. We would also like to thank Bernard Leclerc for pointing out [23], as well as Monica Vazirani and Weiqiang Wang for some useful comments. Finally, we thank the referees for their careful reading of the paper.
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Research of J. R. Kujawa was partially supported by NSF grant DMS-0734226. Research of D. Hill and J. Sussan was partially supported by NSF EMSW21-RTG grant DMS-0354321.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Hill, D., Kujawa, J.R. & Sussan, J. Degenerate affine Hecke–Clifford algebras and type Q Lie superalgebras. Math. Z. 268, 1091–1158 (2011). https://doi.org/10.1007/s00209-010-0712-7
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DOI: https://doi.org/10.1007/s00209-010-0712-7
Keywords
- Spin symmetric groups
- Affine Hecke–Clifford algebras
- Queer Lie superalgebras
- Categorification of quantum groups