Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 020, 21 pages      arXiv:2502.02353      https://doi.org/10.3842/SIGMA.2026.020

Motives of Nullcones of Quiver Representations

Lydia Gösmann and Markus Reineke
Faculty of Mathematics, Ruhr University Bochum, D-44801 Bochum, Germany

Received June 13, 2025, in final form February 10, 2026; Published online March 02, 2026

Abstract
We derive two formulas for motives of nullcones of quiver representations, one being explicit, the other being of wall-crossing type.

Key words: nullcone; quiver representation; motive; Hesselink stratification.

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References

  1. Bozec T., Schiffmann O., Vasserot E., On the number of points of nilpotent quiver varieties over finite fields, Ann. Sci. Éc. Norm. Supér. 53 (2020), 1501-1544, arXiv:1701.01797.
  2. Bridgeland T., An introduction to motivic Hall algebras, Adv. Math. 229 (2012), 102-138, arXiv:1002.4372.
  3. Efimov A.I., Cohomological Hall algebra of a symmetric quiver, Compos. Math. 148 (2012), 1133-1146, arXiv:1103.2736.
  4. Fine N.J., Herstein I.N., The probability that a matrix be nilpotent, Illinois J. Math. 2 (1958), 499-504.
  5. Hesselink W.H., Desingularizations of varieties of nullforms, Invent. Math. 55 (1979), 141-163.
  6. Hilbert D., Ueber die vollen Invariantensysteme, Math. Ann. 42 (1893), 313-373.
  7. Kraft H., Wallach N.R., On the nullcone of representations of reductive groups, Pacific J. Math. 224 (2006), 119-139.
  8. Le Bruyn L., Optimal filtrations on representations of finite-dimensional algebras, Trans. Amer. Math. Soc. 353 (2001), 411-426.
  9. Le Bruyn L., Noncommutative geometry and Cayley-smooth orders, Pure Appl. Math., Vol. 290, Chapman & Hall/CRC, Boca Raton, FL, 2008.
  10. Le Bruyn L., Procesi C., Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), 585-598.
  11. Mozgovoy S., Introduction to Donaldson-Thomas invariants, in Advances in Representation Theory of Algebras, EMS Ser. Congr. Rep., European Mathematical Society, Zürich, 2013, 195-210.
  12. Reineke M., Counting rational points of quiver moduli, Int. Math. Res. Not. 2006 (2006), 70456, 19 pages, arXiv:math.AG/0505389.
  13. Reineke M., Quiver moduli and small desingularizations of some GIT quotients, in Representation Theory – Current Trends and Perspectives, EMS Ser. Congr. Rep., European Mathematical Society, Zürich, 2017, 613-635, arXiv:1511.08316.
  14. Reineke M., Donaldson-Thomas invariants of symmetric quivers, arXiv:2410.03219.

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