Hall polynomials for $2$-nakayama algebras

Document Type : Research Paper

Author

Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, Iran

Abstract

In this note we show that Hall polynomials exist for basic connected right $2$-Nakayama algebras. This result prove the Ringel's conjecture for right $2$-Nakayama algebras.

Keywords


[1] I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebra, Techniques of repre-sentation theory, 1, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006.
[2] J. A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math., 120 (1995) 361–377.
[3] A. Hubery, Hall polynomials for affine quivers, Represent. Theory, 14 (2010) 355–378.
[4] I. G. Macdonald, Symmetric functions and Hall polynomials, Second edition. With contributions by A. Zelevinskyi Oxford Mathematical Monographs. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
[5] A. Nasr-Isfahani, Hall polynomials and composition algebra of representation finite algebras, Algebr. Represent. Theory, 17 (2014) 1155–1161.
[6] A. Nasr-Isfahani, Hall polynomials for Nakayama algebras, Algebr. Represent. Theory, 15 (2012) 483–490.
[7] A. Nasr-Isfahani and M. Shekari, Right n-Nakayama algebras and their representations, Algebr. Represent. Theory, 23 (2020) 1297–1317.
[8] C. M. Ringel, From representation of quivers via Hall and Loewy algebras to quantum groups, Contemp. Math., 131 (1992) 381–401.
[9] C. M. Ringel, Green’s theorem on Hall algebras, Representation theory of algebras and related topics (Mexico City, 1994), 19 (1996) 185–245.
[10] C. M. Ringel, Hall algebras, in Topics in Algebra, part I, Banach Centre Publ., PWN, Warsaw, 26 (1990) 433–447.
[11] C. M. Ringel, Hall algebras and quantum groups, Invent. Math., 101 (1990) 583–591.
[12] C. M. Ringel, Hall polynomials for the representation-finite hereditary algebras, Adv. Math., 84 (1990) 137–178.
[13] C. M. Ringel, Lie algebras arising in representation theory, in, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, UK, 168 (1992) 284–291.
[14] C. M. Ringel, The Theorem of Bo Chen and Hall polynomials, Nagoya Math. J., 183 (2006) 143–160.
[15] B. Sevenhant and M. Van den Bergh, A relation between a conjecture of Kac and the structure of the Hall algebra, J. Pure Appl. Algebra, 160 (2001) 319–332.
[16] S. Zhang, The Hall polynomials for tame quiver algebras, J. Algebra, 239 (2001) 606–614.