Duke Mathematical Journal

Classes of degeneracy loci for quivers: The Thom polynomial point of view

László Fehér and Richárd Rimányi

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The Chern classes for degeneracy loci of quivers are natural generalizations of the Thom-Porteous-Giambelli formula. Suppose that $E,F$ are vector bundles over a manifold $M$ and that $s : E\to F$ is a vector bundle homomorphism. The question is, which cohomology class is defined by the set $\Sigma\sb k(s)\subset M$ consisting of points $m$ where the linear map $s(m)$ has corank $k$? The answer, due to I. Porteous, is a determinant in terms of Chern classes of the bundles $E,F$. We can generalize the question by giving more bundles over $M$ and bundle maps among them. The situation can be conveniently coded by an oriented graph, called a quiver, assigning vertices for bundles and arrows for maps.

We give a new method for calculating Chern class formulae for degeneracy loci of quivers. We show that for representation-finite quivers this is a special case of the problem of calculating Thom polynomials for group actions. This allows us to apply a method for calculating Thom polynomials developed by the authors. The method–reducing the calculations to solving a system of linear equations–is quite different from the method of A. Buch and W. Fulton developed for calculating Chern class formulae for degeneracy loci of $A\sb n$-quivers, and it is more general (can be applied to $A\sb n$-, $D\sb n$-, $E\sb 6$-, $E\sb 7$-, and $E\sb 8$-quivers). We provide sample calculations for $A\sb 3$- and $D\sb 4$-quivers.

Article information

Duke Math. J., Volume 114, Number 2 (2002), 193-213.

First available in Project Euclid: 18 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
Secondary: 14M12: Determinantal varieties [See also 13C40] 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 16G20: Representations of quivers and partially ordered sets


Fehér, László; Rimányi, Richárd. Classes of degeneracy loci for quivers: The Thom polynomial point of view. Duke Math. J. 114 (2002), no. 2, 193--213. doi:10.1215/S0012-7094-02-11421-5. https://projecteuclid.org/euclid.dmj/1087575408

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  • M. Auslander, I. Reiten, and S. O. Smalø, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math. 36, Cambridge Univ. Press, Cambridge, 1995.
  • A. S. Buch and W. Fulton, Chern class formulas for quiver varieties, Invent. Math. 135 (1999), 665--687.
  • G. Bérczi, L. Fehér, and R. Rimányi, Expressions for resultants coming from the global theory of singularities, preprint, 2002, http://math.ohio-state.edu/~rimanyi/cikkek
  • I. N. Bernšteĭn, I. M. Gel'fand, and V. A. Ponomarev, Coxeter functors, and Gabriel's theorem, Uspekhi. Mat. Nauk 28 (1973), 19--33.
  • A. S. Buch, Combinatorics of degeneracy loci, Ph.D. dissertation, University of Chicago, 1999, http://math.mit.edu/~abuch/papers
  • M. Domokos and A. N. Zubkov, Semisimple representations of quivers in characteristic $p$, Algebr. Represent. Theory 5 (2002), 305--317.
  • W. Fulton and P. Pragacz, Schubert Varieties and Degeneracy Loci, Lecture Notes in Math. 1689, Springer, Berlin, 1998.
  • L. Fehér and R. Rimányi, Calculation of Thom polynomials for group actions, preprint, 2000, http://math.ohio-state.edu/~rimanyi/cikkek
  • --------, Schur and Schubert polynomials as Thom polynomials---cohomology of moduli spaces, preprint, 2001, http://math.ohio-state.edu/~rimanyi/cikkek
  • --------, Thom polynomials with integer coefficients, preprint, 2001, http://math.ohio-state.edu/~rimanyi/cikkek
  • P. Gabriel, ``Auslander-Reiten squences and representation-finite algebras'' in Representation Theory (Ottawa, Ontario, 1979), Vol. I, Lecture Notes in Math. 831, Springer, Berlin, 1980, 1--71.
  • M. É. Kazarian, Characteristic classes of Lagrangian and Legendrian singularities (in Russian), Uspekhi Mat. Nauk. 50 (1995), 45--70.; English translation in Russian Math. Surveys 50 (1995), 701--726.
  • --. --. --. --., ``Characteristic classes of singularity theory'' in The Arnold-Gelfand Mathematical Seminars: Geometry and Singularity Theory, Birkhäuser, Boston, 1997, 325--340.
  • L. Le Bruyn and C. Procesi, Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), 585--598.
  • I. Porteous, ``Simple singularities of maps'' in Proceedings of Liverpool Singularities Symposium (1969/1970), Vol. I, Lecture Notes in Math. 192, Springer, Berlin, 1971, 286--307.
  • R. Rimányi, Thom polynomials, symmetries and incidences of singularities, Invent. Math. 143 (2001), 499--521.
  • D. Voigt, Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen, Lecture Notes in Math 592, Springer, Berlin, 1977.