Abstract
Let \(\Bbbk \) be an algebraically closed field, Q a finite quiver, and denote by \(\textup {rep}_{Q}^{\mathbf {d}}\) the affine \(\Bbbk \)-scheme of representations of Q with a fixed dimension vector d. Given a representation M of Q with dimension vector d, the set \({\mathcal {O}}_{M}\) of points in \(\Bbbk \) isomorphic as representations to M is an orbit under an action on \(\textup {rep}^{\mathbf {d}}_{Q}\Bbbk \) of a product of general linear groups. The orbit \({\mathcal {O}}_{M}\) and its Zariski closure \(\overline {\mathcal {O}}_{M}\), considered as reduced subschemes of \(\textup {rep}_{Q}^{{\mathbf {d}}}\), are contained in an affine scheme \({\mathcal {C}}_{M}\) defined by suitable rank conditions associated to M. For all Dynkin and extended Dynkin quivers, the sets of points of \(\overline {{\mathcal {O}}}_{M}\) and \({\mathcal {C}}_{M}\) coincide, or equivalently, \(\overline {{\mathcal {O}}}_{M}\) is the reduced scheme associated to \({\mathcal {C}}_{M}\). Moreover, \(\overline {\mathcal {O}}_{M}={\mathcal {C}}_{M}\) provided Q is a Dynkin quiver of type \({\mathbb {A}}\), and this equality is a conjecture for the remaining Dynkin quivers (of type \(\mathbb {D}\) and \({\mathbb {E}}\)). Let Q be a Dynkin quiver of type \(\mathbb {D}\) and M a finite dimensional representation of Q. We show that the equality \(T_{N}\overline {\mathcal {O}}_{M}=T_{N}{\mathcal {C}}_{M}\) of Zariski tangent spaces holds for any closed point N of \(\overline {\mathcal {O}}_{M}\). As a consequence, we describe the tangent spaces to \(\overline {\mathcal {O}}_{M}\) in representation theoretic terms.
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References
Abeasis, S., Del Fra, A.: Degenerations for the representations of an equioriented quiver of type Am. Boll. Un. Mat. Ital. Suppl. 2, 157–171 (1980)
Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., vol. 36. Cambridge Univ. Press, Cambridge (1995)
Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. vol. 1, London Math. Soc. Stud. Texts, vol. 65, Cambridge Univ. Press, Cambridge (2006)
Bongartz, K.: Degenerations for representations of tame quivers. Ann. Sci. École Norm. Sup. (4) 28(5), 647–668 (1995)
Bongartz, K.: On degenerations and extensions of finite-dimensional modules. Adv. Math. 121(2), 245–287 (1996)
Demazure, M., Gabriel, P.: Introduction to Algebraic Geometry and Algebraic Groups, North-Holland Math. Stud. vol., 39, North-Holland, Amsterdam-New York (1980)
Gabriel, P.: Auslander-Reiten sequences and representation-finite algebras, Representation Theory, I, Lecture Notes in Math., vol. 831, pp 1–71. Springer, Berlin (1980)
Happel, D.: Triangulated Categories in the Representation Theory of Finite-dimensional Algebras, London Math. Soc. Lecture Note Ser., vol. 119. Cambridge Univ. Press, Cambridge (1988)
Hesselink, W.: Singularities in the nilpotent scheme of a classical group. Trans. Amer. Math. Soc. 222, 1–32 (1976)
Lakshmibai, V., Magyar, P.: Degeneracy schemes, quiver schemes, and Schubert varieties. Internat. Math. Res. Notices 12, 627–640 (1998)
Reiten, I., Van den Bergh, M.: Noetherian hereditary abelian categories satisfying Serre duality. J. Amer. Math. Soc. 15(2), 295–366 (2002)
Riedtmann, Ch.: Algebren, Darstellungsköcher, Überlagerungen und zurück. Comment. Math. Helv. 55(2), 199–224 (1980)
Riedtmann, Ch.: Degenerations for representations of quivers with relations. Ann. Sci. École Norm. Sup. (4) 19(2), 275–301 (1986)
Riedtmann, Ch., Zwara, G.: Orbit closures and rank schemes. Comment. Math. Helv. 88(1), 55–84 (2013)
Ringel, C.M.: Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., vol. 1099. Springer, Berlin (1984)
Voigt, D.: Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen, Lecture Notes in Math., vol. 592. Springer, Berlin (1977)
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The both authors gratefully acknowledge the support of the National Science Centre grant no. 2020/37/B/ST1/00127.
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Presented by: Pramod Achar
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Bobiński, G., Zwara, G. Tangent Spaces of Orbit Closures for Representations of Dynkin Quivers of Type \(\mathbb {D}\). Algebr Represent Theor 26, 1951–1974 (2023). https://doi.org/10.1007/s10468-022-10160-3
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DOI: https://doi.org/10.1007/s10468-022-10160-3