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Representation Type of Borel-Schur Algebras

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In our previous work (Erdmann et al. J. Algebra Appl. 17(2),1850028, 2018), we found all Borel-Schur algebras of finite representation type. In the present article, we determine which Borel-Schur algebras of infinite representation type are tame, and which are wild.

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References

  1. Bautista, R., Salmerón, L., Zuazua, R.: Differential Tensor Algebras and Their Module Categories, Volume 362 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  2. Boltje, R., Hartmann, R.: Permutation resolutions for Specht modules. J Algebraic Combin. 34(1), 141–162 (2011)

    Article  MathSciNet  Google Scholar 

  3. Doty, S.R., Erdmann, K., Martin, S., Nakano, D.K.: Representation type of Schur algebras. Math Z. 232(1), 137–182 (1999)

    Article  MathSciNet  Google Scholar 

  4. Drozd, Y.A.: Tame and wild matrix problems. In: Representations and Quadratic Forms (Russian), 154. Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, pp 39–74 (1979)

  5. Erdmann, K.: Blocks of Tame Representation Type and Related Algebras, Volume 1428 of Lecture Notes in Mathematics. Springer, Berlin (1990)

    Google Scholar 

  6. Erdmann, K., Holm, T.: Algebras and Representation Theory. Springer Undergraduate Mathematics Series. Springer, Cham (2018)

    MATH  Google Scholar 

  7. Erdmann, K., Nakano, D.K.: Representation type of q-Schur algebras. Trans. Amer. Math Soc. 353(12), 4729–4756 (2001)

    Article  MathSciNet  Google Scholar 

  8. Erdmann, K., Santana, A.P., Yudin, I.: On Auslander-Reiten sequences for Borel-Schur algebras. J. Algebra Appl. 17(2), 1850028, 28 (2018)

    Article  MathSciNet  Google Scholar 

  9. Gabriel, P.: Unzerlegbare Darstellungen. I. Manuscripta Math 6, 71–103 (1972). correction, ibid. 6 (1972), 309

    Article  MathSciNet  Google Scholar 

  10. Gabriel, P.: Finite representation type is open. In: Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974), Paper No. 10, pages 23 pp. Carleton Math. Lecture Notes, No. 9 (1974)

  11. Gabriel, P.: The universal cover of a representation-finite algebra. In: Representations of Algebras (Puebla, 1980), Volume 903 of Lecture Notes in Math, pp 68–105. Springer, Berlin (1981)

  12. Geiss, C.: On degenerations of tame and wild algebras. Arch. Math. (Basel) 64(1), 11–16 (1995)

    Article  MathSciNet  Google Scholar 

  13. Green, J.A.: On certain subalgebras of the Schur algebra. J. Algebra 131(1), 265–280 (1990)

    Article  MathSciNet  Google Scholar 

  14. Green, J.A.: Polynomial Representations of GLn, volume 830 of Lecture Notes in Mathematics. Springer, Berlin (2007). Augmented Edition, With an Appendix on Schensted Correspondence and Littelmann Paths by K. Erdmann, Green and M. Schocker

    Google Scholar 

  15. Nazarova, L.A.: Partially ordered sets of infinite type. Izv. Akad. Nauk SSSR Ser. Mat. 39(5), 963–991, 1219 (1975)

    MathSciNet  Google Scholar 

  16. Ringel, C.M.: The indecomposable representations of the dihedral 2-groups. Math. Ann. 214, 19–34 (1975)

    Article  MathSciNet  Google Scholar 

  17. Ringel, C.M.: On algorithms for solving vector space problems. II. tame algebras. In: Representation Theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), volume 831 of Lecture Notes in Math, pp 137–287. Springer, Berlin (1980)

  18. Ringel, C.M.: Tame Algebras and Integral Quadratic Forms, Volume 1099 of Lecture Notes in Mathematics. Springer, Berlin (1984)

    Google Scholar 

  19. Santana, A.P.: The Schur algebra S(B+) and projective resolutions of Weyl modules. J. Algebra 161(2), 480–504 (1993)

    Article  MathSciNet  Google Scholar 

  20. Santana, A.P., Yudin, I.: Characteristic-free resolutions of Weyl and Specht modules. Adv. Math. 229(4), 2578–2601 (2012)

    Article  MathSciNet  Google Scholar 

  21. Wald, B., Waschbüsch, J.: Tame biserial algebras. J. Algebra 95(2), 480–500 (1985)

    Article  MathSciNet  Google Scholar 

  22. Woodcock, D.: Schur algebras and global bases: new proofs of old vanishing theorems. J. Algebra 191(1), 331–370 (1997)

    Article  MathSciNet  Google Scholar 

  23. Woodcock, D.J.: Borel Schur algebras. Comm. Algebra 22(5), 1703–1721 (1994)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. This research was also supported by the program ’Research in Pairs’ by the Mathematical Forschungsinstitut Oberwolfach in 2016. The third author’s position is financed by FCT via CEECIND/04092/2017.

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Correspondence to Ivan Yudin.

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Presented by: Michela Varagnolo

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Erdmann, K., Santana, A.P. & Yudin, I. Representation Type of Borel-Schur Algebras. Algebr Represent Theor 24, 1387–1413 (2021). https://doi.org/10.1007/s10468-020-09995-5

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