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An Introduction to Quiver Representations

About this Title

Harm Derksen, University of Michigan, Ann Arbor, MI and Jerzy Weyman, University of Connecticut, Storrs, CT

Publication: Graduate Studies in Mathematics
Publication Year: 2017; Volume 184
ISBNs: 978-1-4704-2556-2 (print); 978-1-4704-4260-6 (online)
DOI: https://doi.org/10.1090/gsm/184
MathSciNet review: MR3727119
MSC: Primary 16-02; Secondary 13A50, 14L24, 16G10, 16G20, 16G70

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Front/Back Matter

Chapters

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References
  • Abeasis, S., Del Fra, A., Kraft, H., The geometry of representations of $A_m$, Math. Ann. 256 (1981), 401–418.
  • Assem, I., Brüstle, T., Schiffler, R., Cluster-tilted algebras as trivial extensions, Bull. London Math. Soc., 40 (2008), 151–162.
  • Auslander, M., Reiten, I., Smalø, S., Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, New York, 1995.
  • Auslander, M., Reiten, I., Representation theory of Artin algebras III, Almost split sequences, Comm. in Algebra, 3 (1975), 269–310.
  • Auslander, M., Reiten, I., Representation theory of Artin algebras IV, Invariants given by almost split sequences, Comm. in Algebra , 5 (1977), 443–518.
  • Auslander, M., Reiten, I., Modules determined by their composition factors, Ill. Journal of Math., 29 (1985), 280–301.
  • Belkale, P., Geometric proof of a conjecture by Fulton, Adv. in Math., 216 (2007), 346–357.
  • Bernstein, I.N., Gelfand, I.M., Ponomarev, V.A., Coxeter functors and Gabriel’s theorem, Russian Math. Surveys, 28 (1973), 17–32.
  • Bobiński, G., Zwara, G., Normality of orbit closures for Dynkin quivers of type $A_n$, Manuscripta Math., 105 (2001), no. 1, 103–109.
  • Bobiński, G., Zwara, G., Schubert varieties and representations of Dynkin quivers, Colloq. Math., 94 (2002), no. 2, 285–309.
  • Brion, M., Multiplicity-free sub varieties of flag varieties, Commutative Algebra (Grenoble/Lyon, 2001), 13-23, Contemp. Math 331, Amer. Math. Soc., Providence, RI, 2003.
  • Buan, A.B., Marsh, R., Reineke, M., Reiten, I., Todorov, G., Tilting theory and cluster combinatorics, Adv. in Math., 204 (2006), no. 2, 572–618.
  • Buan, A.B., Marsh, R., Reiten, I., Cluster-tilted algebras, Trans. Amer. Math. Soc., 359 (2007), 323–332.
  • Caldero, Ph., Chapoton, F., Schiffler, R., Quivers with relations arising from clusters ($A_n$ case), Trans. Amer. Math.Soc., 358, no. 3, (2006), 359–376.
  • Chindris, C., Derksen, H., Weyman, J., Counterexamples to Okounkov’s log-concavity conjecture, Compos. Math. 143 (2007), no. 6, 1545–1557.
  • Crawley-Boevey, W., Exceptional sequences of representations of quivers, Representations of algebras (Ottawa, ON 1992), CMS Conf. Proc., 14, Amer. Math. Soc., Providence, RI, 1993.
  • Crawley-Boevey, W., Subrepresentations of general representations of quivers, Bull. London Math. Soc., 28 (1996), 363–366.
  • Crawley-Boevey, W., Lectures on Representations of Quivers, preprint available from Crawley-Boevey’s home page.
  • Crawley-Boevey, W., Holland, M.P., Non-commutative deformations of Kleinian singularities, Duke Math. J., 92 (1998), 605–635.
  • Crawley-Boevey, W., Van den Bergh, M., Absolutely indecomposable representations and Kac-Moody Lie algebras, Invent. Math. 155 (2004), no. 3, 537–559.
  • Carroll, A., Weyman, J., Noncommutative Birational Geometry, Representations and Combinatorics, "Contemporary Mathematics" , Proceeding of the Special AMS session "Noncommutative birational geometry and cluster algebras" held during the Annual meeting of the AMS (Boston, 2012), edited by A. Berenstein, V. Retakh, 111–136, AMS, 2013.
  • Derksen, H., Fei, J., General presentations of algebras, Adv. Math. 278 (2015), 210–237.
  • Derksen, H., Makam, V., Polynomial degree bounds for matrix semi-invariants, Adv. Math. 310 (2017), 44-63.
  • Derksen, H, Makam, V., Generating invariant rings of quivers in arbitrary characteristic, J. Algebra 489 (2017), 435–445.
  • Derksen, H. Weyman, J. Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, J. Amer. Math. Soc. 13 (2000), no. 3, 467–479.
  • Derksen, H., Weyman, J., On the canonical decomposition of quiver representations, Compositio Math. 133 (2002), no. 3, 245–256.
  • Derksen, H. Weyman, J., On the Littlewood-Richardson polynomials, J. Algebra 255 (2002), no. 2, 247–257.
  • Derksen, H., Weyman, J., Semi-invariants for quivers with relations, Special issue in celebration of Claudio Procesi’s 60-th birthday, J. Algebra 258 (2002), no. 1, 216–227.
  • Derksen, H., Weyman, J., Generalized quivers associated to reductive groups, Colloq. Math. 94 (2002), no. 2, 151–173.
  • Derksen, H., Weyman, J., The combinatorics of quiver representations, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 3, 1061–1131.
  • Derksen, H. Schofield, A., Weyman, J., On the number of sub representations of a general quiver representation, J. London. Math. Sco. (2) 76 (2007), no. 1, 135–147.
  • Derksen, H., Weyman, J., Zelevinsky, A., Quivers with potential and their representations I. Mutations, Selecta Math. (N.S.) 14 (2008), no. 1, 59–119.
  • Derksen, H., Weyman, J., Zelevinsky, A., Quivers with potential and their representations II. applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), no. 3, 749–790.
  • Dlab, V., Ringel, M., Indecomposable Representations of Graphs and Algebras, Mem. Amer. Math. Soc., 173, 1976.
  • Domokos, M., Relative invariants for representations of finite dimensional algebras, Manuscripta Math., 108 (2002), 123–133.
  • Domokos, M., Zubkov, A.N., Semi-invariants of quivers as determinants, Transformation Groups 6 (2001), No. 1, 9-24.
  • Donovan, P., Freislich, M.R., The Representation Theory of Finite Graphs and Associative Algebras, Carleton Lecture Notes, Ottawa 1973.
  • Assem, I., Simson, D., Skowroński, A., Elements of the Representation Theory of Associative Algebras, London Mathematical Society, Student Texts 65, Cambridge University Press, 2006.
  • Fei, J., Cluster algebras, invariant theory, and Kronecker coefficients I, Adv. Math. 310 (2017), 1064–1112.
  • Fei, J. Cluster algebras and semi-invariant rings II: projections, Math. Z. 285 (2017), no. 3–4, 939–966.
  • Fei, J., The upper cluster algebras of IART quivers I. Dynkin, arXiv 1603.02521.
  • Fomin, S., Shapiro, M., Thurston, D., Cluster algebras and triangulated surfaces, I. Cluster complexes, Acta Math., 201 (2008), no 1, 83–146.
  • Gabriel, P., Roiter, A.V., Representations of finite dimensional algebras, Encyclopaedia of Mathematical Sciences, no. 73, Springer-Verlag, 2nd edition, 1997.
  • Gelfand, S.I., Manin, Y.I., Methods of Homological Algebra, Springer Monographs in Mathematics, Springer, 2003.
  • Goodman, R., Wallach, N.R., Representations and Invariants of Classical Groups, Cambridge University Press, 1998,
  • Gorodentsev, A.L., Rudakov, A.N., Exceptional vector bundles on projective spaces, Duke Math. J., 54 (1987), no. 1, 115–130.
  • Happel, Triangulated categories in the Representation Theory of Finite Dimensional Algebras, London Mathematical Society Lecture Notes Series 119, Cambridge University Press, 1988.
  • Hausel, T, Letellier, E., Rodrigues-Villegas, F., Positivity of Kac polynomials and DT-invariants of quivers, Annals of Mathematics (2) 177 (2013), no. 3, 1147–1168.
  • Hille, L., On the volume of a tilting module, Abh. Math. Sem. Univ. Hamburg 76 (2006), 261–277.
  • Humphreys, J., Linear Algebraic Groups, Graduate Texts in Mahematics, vol. 21, Springer, 1975.
  • Igusa, K., Orr, K., Todorov, G., Weyman, J., Cluster complexes via semi-invariants, Compositio Mathematica 145, no. 4 (2009), 1001–1034.
  • Ikenmeyer, Ch., Small Littlewood-Richardson coefficients, J. Algebraic Comb. 44 (2016), no. 1, 1–29; Erratum, J. Algebraic Comb. 44 (2016), no. 1, 31–32.
  • Kac, V., Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), no. 1, 57–92.
  • Kac, V., Infinite root systems, representations of graphs and invariant theory II, J. Algebra 78 (1982), no. 1, 141–162.
  • Kac, V., Root systems, representations of quivers and invariant theory, in: Invariant Theory (Montecatini 1982), Lecture Notes in Mathematics, 996, Springer, Berlin, 1983).
  • Keller, B., Triangulated orbit categories, Documenta Mathematica 10 (2005), 551-581, arXiv 0503240.
  • King, A.D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser (2) 45 (1994), no, 180, 515–530.
  • King, R., Tollu, Ch., Toumazet, F., Stretched Littlewood-Richardson polynomials and Kostka coefficients, CRM Proceedings and Lecture Notes 34 (2004), 99–112.
  • Kleiner, M., The graded preprojective algebra of a quiver, Bull. London Math. Soc. 36 (2004), no. 1, 13–22.
  • Kraft, H., Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1., Fried. Viehweg $\&$ Sohn, Braunschweig, 1984.
  • Lakshmibai, V., Magyar, P., Degeneracy schemes, quiver schemes, and Schubert varieties, Internat. Math. Res. Notices 1998, no. 12, 627–640.
  • Le Bruyn, L., Procesi, C., Semisimple representations of quivers, Trans. Amer. Mah.Soc. 317 (1990), no. 2, 585–598.
  • MacDonald, I.G., Symmetric functions and Hall polynomials, 2nd edition, Oxford Mathematical Monographs, Oxford University Press, 1995.
  • Marsh, R., Reineke, M., Zelevinsky, A., Generalized associahedra via quiver representations, Trans. Amer. Math. Soc, 355 (2003), 4171–4186.
  • Mumford, D., Fogarty, J., Kirwan, F., Geometric Invariant Theory. Third edition. Results in Mathematics and Related res (2), 34, Springer-Verlag, Berlin, 1994.
  • Nazarova, L.A., Representations of quivers of infinite type, Izv. Akad. Nauk SSSR, Ser. Mat. 37 (1973), 752–791.
  • Obaid, M., Nauman, K., Al-Shammakh, W.S.M., Fakieh, W., Ringel, C.M., The number of complete exceptional sequences for a Dynkin algebra, Colloq. Math. 133 (2013), no. 2, 197–210.
  • Ringel, C.M., The braid group action on the set of exceptional sequences of a hereditary Artin algebra, in: Contemporary Mathematics volume 171, 1994, 339–352.
  • Ringel, C.M., Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics, 1099, 1984.
  • Rotman, J., An Introduction to Homological Algebra, 2-nd edition, Universitext, No. 223, Springer, 2009.
  • Sato, M., Kimura, T., A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155.
  • Schofield, A., Semi-invariants of quivers, J. London Math. Soc. (2) (1991), no. 3, 385–395.
  • Schofield, A., General representations of quivers, Proc. London Math. Soc. (3) 65 (1992), no. 1, 46–64.
  • Schofield, A., The internal structure of real Schur representations, preprint, 1988.
  • Schofield, A., Van den Bergh, M., Semi-invariants of quivers for arbitrary dimension vectors, Indag. Math. (N.S.) 12 (2001), no. 1, 125–138.
  • Sherman, C., Quiver generalization of a conjecture of King, Tollu and Toumazet, J. Algebra 480 (2017), 487–504.
  • Skowroński, A., Weyman, J., The algebras of semi-invariants of quivers, Transform. Groups 5 (2000), no. 4, 361–402.
  • Zwara, G., Smooth morphisms of module schemes, Proc. London Math. Soc., (3) 84 (2002), no. 3, 539–558.