# The incidence class and the hierarchy of orbits

Open Mathematics (2009)

• Volume: 7, Issue: 3, page 429-441
• ISSN: 2391-5455

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## Abstract

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R. Rimányi defined the incidence class of two singularities η and ζ as [η]|ζ, the restriction of the Thom polynomial of η to ζ. He conjectured that (under mild conditions) [η]|ζ ≠ 0 ⇔ ζ ⊂ $\overline{\eta }$ . Generalizing this notion we define the incidence class of two orbits η and ζ of a representation. We give a sufficient condition (positivity) for ζ to have the property that [η]|ζ ≠ 0 ⇔ ζ ⊂ $\overline{\eta }$ for any other orbit η. We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity holds for all orbits. In other words in these cases the incidence classes completely determine the hierarchy of the orbits. We also study the case of singularities where positivity doesn’t hold for all orbits.

## How to cite

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László Fehér, and Zsolt Patakfalvi. "The incidence class and the hierarchy of orbits." Open Mathematics 7.3 (2009): 429-441. <http://eudml.org/doc/269032>.

@article{LászlóFehér2009,
abstract = {R. Rimányi defined the incidence class of two singularities η and ζ as [η]|ζ, the restriction of the Thom polynomial of η to ζ. He conjectured that (under mild conditions) [η]|ζ ≠ 0 ⇔ ζ ⊂ $\bar\{\eta \}$ . Generalizing this notion we define the incidence class of two orbits η and ζ of a representation. We give a sufficient condition (positivity) for ζ to have the property that [η]|ζ ≠ 0 ⇔ ζ ⊂ $\bar\{\eta \}$ for any other orbit η. We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity holds for all orbits. In other words in these cases the incidence classes completely determine the hierarchy of the orbits. We also study the case of singularities where positivity doesn’t hold for all orbits.},
author = {László Fehér, Zsolt Patakfalvi},
journal = {Open Mathematics},
keywords = {Thom polynomial; Equivariant maps; Equivariant Poincaré dual; Multidegree; Joseph polynomial; Incidence class; equivariant maps; equivariant Poincaré dual; multidegree; incidence class},
language = {eng},
number = {3},
pages = {429-441},
title = {The incidence class and the hierarchy of orbits},
url = {http://eudml.org/doc/269032},
volume = {7},
year = {2009},
}

TY - JOUR
AU - László Fehér
AU - Zsolt Patakfalvi
TI - The incidence class and the hierarchy of orbits
JO - Open Mathematics
PY - 2009
VL - 7
IS - 3
SP - 429
EP - 441
AB - R. Rimányi defined the incidence class of two singularities η and ζ as [η]|ζ, the restriction of the Thom polynomial of η to ζ. He conjectured that (under mild conditions) [η]|ζ ≠ 0 ⇔ ζ ⊂ $\bar{\eta }$ . Generalizing this notion we define the incidence class of two orbits η and ζ of a representation. We give a sufficient condition (positivity) for ζ to have the property that [η]|ζ ≠ 0 ⇔ ζ ⊂ $\bar{\eta }$ for any other orbit η. We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity holds for all orbits. In other words in these cases the incidence classes completely determine the hierarchy of the orbits. We also study the case of singularities where positivity doesn’t hold for all orbits.
LA - eng
KW - Thom polynomial; Equivariant maps; Equivariant Poincaré dual; Multidegree; Joseph polynomial; Incidence class; equivariant maps; equivariant Poincaré dual; multidegree; incidence class
UR - http://eudml.org/doc/269032
ER -

## References

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