## Algebraic & Geometric Topology

### Representation theory for the Križ model

#### Abstract

The natural action of the symmetric group on the configuration spaces $F(X,n)$ induces an action on the Križ model $E(X,n)$. The representation theory for this complex is studied and a big acyclic subcomplex which is $Sn$–invariant is described.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 1 (2014), 57-90.

Dates
Revised: 4 July 2013
Accepted: 4 July 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715794

Digital Object Identifier
doi:10.2140/agt.2014.14.57

Mathematical Reviews number (MathSciNet)
MR3158753

Zentralblatt MATH identifier
1281.55015

#### Citation

Ashraf, Samia; Azam, Haniya; Berceanu, Barbu. Representation theory for the Križ model. Algebr. Geom. Topol. 14 (2014), no. 1, 57--90. doi:10.2140/agt.2014.14.57. https://projecteuclid.org/euclid.agt/1513715794

#### References

• V I Arnold, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969) 227–231
• S Ashraf, H Azam, B Berceanu, Representation stability of power sets and square free polynomials
• S Ashraf, B Berceanu, Cohomology of $3$–points configuration spaces of complex projective spaces
• S Ashraf, B Berceanu, Equivariant Lefschetz structure of the Križ model, in preparation
• H Azam, B Berceanu, Cohomology of configuration spaces of Riemann surfaces, in preparation
• B Berceanu, M Markl, S Papadima, Multiplicative models for configuration spaces of algebraic varieties, Topology 44 (2005) 415–440
• G M Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978) 178–218
• R Bezrukavnikov, Koszul DG–algebras arising from configuration spaces, Geom. Funct. Anal. 4 (1994) 119–135
• T Church, B Farb, Representation theory and homological stability, Adv. Math. 245 (2013) 250–314
• F R Cohen, L R Taylor, Computations of Gel${}^\prime$fand–Fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, from: “Geometric applications of homotopy theory, I”, (M G Barratt, M E Mahowald, editors), Lecture Notes in Math. 657, Springer, Berlin (1978) 106–143
• F R Cohen, L R Taylor, Configuration spaces: Applications to Gelfand–Fuks cohomology, Bull. Amer. Math. Soc. 84 (1978) 134–136
• P Deligne, P Griffiths, J Morgan, D Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975) 245–274
• E M Feichtner, G M Ziegler, The integral cohomology algebras of ordered configuration spaces of spheres, Doc. Math. 5 (2000) 115–139
• W Fulton, J Harris, Representation theory, Graduate Texts in Mathematics 129, Springer, New York (1991)
• W Fulton, R MacPherson, A compactification of configuration spaces, Ann. of Math. 139 (1994) 183–225
• I Križ, On the rational homotopy type of configuration spaces, Ann. of Math. 139 (1994) 227–237
• P Lambrechts, D Stanley, A remarkable DG module model for configuration spaces, Algebr. Geom. Topol. 8 (2008) 1191–1222
• W Ledermann, Introduction to group characters, 2nd edition, Cambridge Univ. Press (1987)
• G I Lehrer, L Solomon, On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes, J. Algebra 104 (1986) 410–424
• P Orlik, H Terao, Arrangements of hyperplanes, Grundl. Math. Wissen. 300, Springer, Berlin (1992)
• J-P Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer, New York (1977)
• R P Stanley, Some aspects of groups acting on finite posets, J. Combin. Theory Ser. A 32 (1982) 132–161