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Locally free representations of quivers over commutative Frobenius algebras

  • Tamás Hausel [1] ; Emmanuel Letellie [2] ; Fernando Rodriguez-Villega [3]
    1. [1] IST Austria, Klosterneuburg, Austria
    2. [2] Université Paris Cité, France
    3. [3] ICTP Trieste, Trieste, Italy
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  • Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 2, 2024, 50 págs.
  • Idioma: inglés
  • DOI: 10.1007/s00029-023-00914-2
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  • Resumen

    • In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t2). Using these results together with results of Geiss, Leclerc and Schröer we give, when k is algebraically closed, a classification of pairs (Q,R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra Fq [t]/(t r). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.

  • Referencias bibliográficas
    • Bosch S., Lütkebohmert W., Raynaud M.: Néron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related...
    • Bourbaki, N.: Eléments of Mathematics, Algebra I, Chap. II, Linear Algebra
    • Brion, M.: Representations of quivers. Geometric methods in representation theory. I, 103–144, Sémin. Congr., 24-I, Soc. Math. France, Paris...
    • Brown, W.: Matrices over commutative rings. Monographs and Textbooks in Pure and Applied Mathematics, 169. Marcel Dekker, Inc., New York (1993)
    • Carlitz, L.: A combinatorial property of q-Eulerian numbers. Am. Math. Mon. 82, 51–54 (1975)
    • Crawley-Boevey, W.: (Private communication)
    • Deligne, P.: Cohomologie étale, SGA 4-1/2 IV, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, pp. 233–261 (1977)
    • Deligne, P.: La conjecture de Weil?: II. Publications Math. IHES 52, 137–252 (1980)
    • Demazure, M., Gabriel, P.: Groupes algébriques. North-Holland Publishing Co., Amsterdam (1970)
    • Dlab, V., Ringel, C.M.: Indecomposable representations of graphs and algebras. Mem. Am. Math. Soc. 6, 173 (1976)
    • Eilenberg, S., Nakayama, T.: On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings. Nagoya Math. J. 9,...
    • Eisenbud, D.: Commutative algebra. With a view toward algebraic geometry. GTM, Vol. 150. Springer-Verlag, New York (1995)
    • Ellis-Monaghan, J.A., Merino, C.: Graph Polynomials and Their Applications I: The Tutte Polynomial Structural Analysis of Complex Networks,...
    • Gabriel P.: Unzerlegbare Darstellungen. I. Manuscripta Math. 6, 71–103 (1972); correction, ibid. 6 (1972), 309
    • Gabriel, P.: Indecomposable representations. II. Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971), pp. 81–104....
    • Geuenich J.: Quiver Modulations and Potentials, PhD thesis, University of Bonn (2017)
    • Geiss, C., Leclerc, B., Schröer, J.: Semicanonical bases and preprojective algebras. Ann. Sci. École Norm. Sup. (4) 38(2), 193–253 (2005)
    • Geiss, C., Leclerc, B., Schröer, J.: Quivers with relations for symmetrizable matrices I: Foundations. Invent. Math. 209, 61–158 (2017)
    • Geiss, C., Leclerc, B., Schröer, J.: (Private communication)
    • Hazewinkel, M., Gubareni Nn, Kirichenko, V.V.: Algebras, rings and modules, Vol. 2. Mathematics and Its Applications, vol. 586. Springer
    • Hausel, T.: Kac conjecture from Nakajima quiver varieties. Invent. Math. 181, 21–37 (2010)
    • Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Arithmetic harmonic analysis on character and quiver varieties. Duke Math. J. 160(2), 323–400...
    • Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Positivity for Kac polynomials and DT-invariants of quivers. Ann. Math. (2) 177(3), 1147–1168...
    • Hausel, T., Rodriguez-Villegas, F.: Mixed Hodge polynomials of character varieties, With an appendix by Nicholas M. Katz. Invent. Math. 174,...
    • Hua, J.: Counting representations of quivers over finite fields. J. Algebra 226(2), 1011–1033 (2000)
    • Jambor, S., Plesken, W.: Normal forms for matrices over uniserial rings of length two. J. Algebra 358, 25–256 (2012)
    • Kac, V.: Root systems, representations of quivers and invariant theory, Invariant theory (Montecatini, 1982), pp. 74–108. Lecture Notes in...
    • Krajewski, T., Moffatt, I., Tanasa, A.: Hopf algebras and Tutte polynomials Adv. Appl. Math. 95, 271–330 (2018)
    • Kraft H., Riedtmann, Ch.: Geometry of representations of quivers, Representations of algebras (Durham, 1985), pp. 109–145. London Math. Soc....
    • Kaplan, D.: Frobenius degenerations of preprojective algebras. J. Noncommu. Geom. 14(1), 34–411 (2020)
    • Letellier, E.: DT-invariants of quivers and the Steinberg character of gln. IMRN 22, 11887–11908 (2015)
    • Li, F., Ye, Ch.: Representations of Frobenius-type triangular matrix algebras. Acta Math. Sin. 33(3), 341–361 (2017)
    • Macdonald, I.G.: Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, 2nd edn. Oxford Science Publications. The Clarendon...
    • Manchon, D.: Hopf algebras, from basics to applications to renormalization Comptes Rendus des Rencontres Mathematiques de Glanon 2001 (2003)....
    • Milne, J. S.: Algebraic groups. The theory of group schemes of finite type over a field. Cambridge Studies in Advanced Mathematics, 170. Cambridge...
    • Mozgovoy, S.: Motivic Donaldson–Thomas invariants and McKay correspondence. arXiv:1107.6044
    • Prasad, A., Singla, P., Spallone, S.: Similarity of matrices over local rings of length two. Indiana Univ. Math. J. 64, 471–514 (2015)
    • Reiner I.: Maximal orders. 1975 Oxford University Press. http://www.ams.org/mathscinet-getitem?mr=1972204
    • Ringel, C.M., Zhang, P.: Representations of quivers over the algebra of dual numbers. J. Algebra 475, 327–360 (2017)
    • Ringel, C.M., Zhang, P.: From submodule categories to preprojective algebras. Math. Z. 278(1–2), 55–73 (2014)
    • Schiffmann, O.: https://math.stackexchange.com/questions/606279/how-many-pairs-of-nilpotent-commuting-matrices-are-there-in-m-n-mathbbf-q
    • Skowro´nski, A.: Tame triangular matrix algebras over Nakayama algebras. J. Lond. Math. Soc. (2) 34(2), 245–264 (1986)
    • Springer, T.A.: Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. Math. 36, 173–207 (1976)
    • Stanley R.: Enumerative combinatorics, Vol. 2. (English summary) With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. Cambridge...
    • Wyss, D.: Motivic and p-adic Localization Phenomena. arXiv:1709.06769v1
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