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Degenerate 0-Schur Algebras and Nil-Temperley-Lieb Algebras

  • Bernt Tore Jensen
  • Xiuping SuEmail author
  • Guiyu Yang
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Abstract

In Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014) constructed 0-Schur algebras, using double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver presentation naturally gives rise to a new class of algebras, which are introduced and studied in this paper. That is, these algebras are defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters t. In particular, when all the entries of t are 1, we recover 0-Schur algebras. When all the entries of t are zero, we obtain a class of basic algebras, which we call the degenerate 0-Schur algebras. We prove that the degenerate algebras are both associated graded algebras and quotients of 0-Schur algebras. Moreover, we give a geometric interpretation of the degenerate algebras using double flag varieties, in the same spirit as Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014), and show how the centralizer algebras are related to nil-Hecke and nil-Temperley-Lieb algebras.

Keywords

0-Schur algebras Quivers Nil-Hecke algebras Nil-Temperley-Lieb algebras Double flag varieties 

Notes

Acknowledgments

We would like to thank the referee for helpful comments, which have improved the exposition in this paper, and especially for pointing out the reference [15] and it’s relevance to our results in Section 6.

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© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, NTNU in GjøvikNorwegian University of Science and TechnologyGjøvikNorway
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK
  3. 3.School of Mathematics and StatisticsShandong University of TechnologyZiboChina

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