Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-18T11:02:56.162Z Has data issue: false hasContentIssue false
  • English
  • Français

The Theorem of Bo Chen and Hall Polynomials

Published online by Cambridge University Press:  11 January 2016

Claus Michael Ringel
Claus Michael Ringel*
Affiliation:
Fakultät für MathematikUniversität BielefeldPOBox 100 131 D-33 501 BielefeldGermanyringel@math.uni-bielefeld.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Λ be the path algebra of a Dynkin quiver. A recent result of Bo Chen asserts that Hom(X; Y/X) = 0 for any Gabriel-Roiter inclusion X ⊆ Y. The aim of the present note is to give an interpretation of this result in terms of Hall polynomials, and to extend it in this way to representation-directed split algebras. We further show its relevance when dealing with arbitrary representation-finite split algebras.

Keywords

16G2016G6017B3781R50
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 183 , 2006 , pp. 143 - 160
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

References

[B] Bongartz, K., Indecomposables are standard, Comment. Math. Helv., 60 (1985), 400–410.CrossRefGoogle Scholar
[BG] Bongartz, K. and Gabriel, P., Covering spaces in representation theory, Invent. Math., 65 (1982), 331–378.Google Scholar
[C] Chen, B., The Gabriel-Roiter measure for representation-finite hereditary algebras, Dissertation, Bielefeld 2006.Google Scholar
[CB] Crawley-Boevey, W., Rigid integral representations of quivers, Can. Math. Soc. Conf. Proc. 18 (1996), pp. 155–163.Google Scholar
[G] Gabriel, P., Indecomposable representations II, Symposia Mat. Inst. Naz. Alta Mat., 11 (1973), 81–104. MR 49:5132.Google Scholar
[L] Lusztig, G., Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc., 3 (1990), 447–498.Google Scholar
[M] Macdonald, I. G., Symmetric functions and Hall polynomials, second edition, Oxford Math. Mon., 1995.Google Scholar
[R1] Ringel, C. M., Tame algebras and integral quadratic forms, Springer LNM 1099, 1984.Google Scholar
[R2] Ringel, C. M., Hall algebras, Topics in Algebra, Banach Center Publ. 26 (1990), pp. 433–447.Google Scholar
[R3] Ringel, C. M., Hall polynomials for the representation-finite hereditary algebras, Adv. Math., 84 (1990), 137–178.CrossRefGoogle Scholar
[R4] Ringel, C. M., Exceptional objects in hereditary categories, Proceedings Constantza Conference, An. St. Univ. Ovidius Constantza Vol. 4 (1996), f. 2, 150–158.Google Scholar
[R5] Ringel, C. M., The braid group action on the set of exceptional sequences of a hereditary algebra, Abelian Group Theory and Related Topics, Contemp. Math. 171 (1994), pp. 339–352.Google Scholar
[R6] Ringel, C. M., The Gabriel-Roiter measure, Bull. Sci. math., 129 (2005), 726–748.CrossRefGoogle Scholar
[R7] Ringel, C. M., Foundation of the Representation Theory of Artin Algebras, Using the Gabriel-Roiter Measure, Proceedings ICRA 11, Queretaro 2004, Contemporary Math. 406, Amer. Math. Soc. (2006), pp. 105–135.Google Scholar
[S] Schofield, A., The internal structure of real Schur representations, preprint, 46–64.Google Scholar