Abstract
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.
Similar content being viewed by others
References
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)
Boltje, R., Hartmann, R.: Permutation resolutions for Specht modules. J Algebraic Combin. 34(1), 141–162 (2011)
Doty, S.R., Erdmann, K., Martin, S., Nakano, D.K.: Representation type of Schur algebras. Math Z. 232(1), 137–182 (1999)
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)
Erdmann, K.: Blocks of Tame Representation Type and Related Algebras, Volume 1428 of Lecture Notes in Mathematics. Springer, Berlin (1990)
Erdmann, K., Holm, T.: Algebras and Representation Theory. Springer Undergraduate Mathematics Series. Springer, Cham (2018)
Erdmann, K., Nakano, D.K.: Representation type of q-Schur algebras. Trans. Amer. Math Soc. 353(12), 4729–4756 (2001)
Erdmann, K., Santana, A.P., Yudin, I.: On Auslander-Reiten sequences for Borel-Schur algebras. J. Algebra Appl. 17(2), 1850028, 28 (2018)
Gabriel, P.: Unzerlegbare Darstellungen. I. Manuscripta Math 6, 71–103 (1972). correction, ibid. 6 (1972), 309
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)
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)
Geiss, C.: On degenerations of tame and wild algebras. Arch. Math. (Basel) 64(1), 11–16 (1995)
Green, J.A.: On certain subalgebras of the Schur algebra. J. Algebra 131(1), 265–280 (1990)
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
Nazarova, L.A.: Partially ordered sets of infinite type. Izv. Akad. Nauk SSSR Ser. Mat. 39(5), 963–991, 1219 (1975)
Ringel, C.M.: The indecomposable representations of the dihedral 2-groups. Math. Ann. 214, 19–34 (1975)
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)
Ringel, C.M.: Tame Algebras and Integral Quadratic Forms, Volume 1099 of Lecture Notes in Mathematics. Springer, Berlin (1984)
Santana, A.P.: The Schur algebra S(B+) and projective resolutions of Weyl modules. J. Algebra 161(2), 480–504 (1993)
Santana, A.P., Yudin, I.: Characteristic-free resolutions of Weyl and Specht modules. Adv. Math. 229(4), 2578–2601 (2012)
Wald, B., Waschbüsch, J.: Tame biserial algebras. J. Algebra 95(2), 480–500 (1985)
Woodcock, D.: Schur algebras and global bases: new proofs of old vanishing theorems. J. Algebra 191(1), 331–370 (1997)
Woodcock, D.J.: Borel Schur algebras. Comm. Algebra 22(5), 1703–1721 (1994)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Michela Varagnolo
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-020-09995-5
Keywords
- Borel-Schur algebras
- Representation type
- Coverings of algebras
- Auslander-Reiten quivers
- Representations of posets