The aim of the Spring School is to work through the proof, by T. Hausel, E. Letellier and F. RodriguezVillegas, of the Kac conjectures on numbers of indecomposable representations of quivers, thereby covering the methods and techniques necessary for the proof, such as the representation theory of quivers, the geometry of quiver representations, Nakajima quiver varieties and their cohomology, the character theory of finite groups of Lie type, etc..
The Spring School is intended for Ph.D. students and (early) postdocs working on or interested in these fields.
The Spring School will follow the tradition of previous spring/summer schools in representation theory. Every participant chooses a subject of the program (see below) and gives a talk. The typical participant will not be an expert in the subject, but working in one of the fields indicated above and interested in jointly learning a new subject. There will be no special prerequisities except for standard knowledge in algebra, but participants are expected to prepare themselves and their talks in advance. It is recommended that this is done in small groups; in particular, the talks could/should be subdivided into smaller parts.
At the end of the week, there will be lectures by S. Mozgovoy (Dublin) on more recent developments in the field.
How deeply we will go into the technical details of the proofs will depend on the interests and bachgrounds of the participants. Therefore, at this moment we will only give a rough outline of the programme; the details of what material each talk should contain will be posted later when the list of participants is fixed.
Define quivers and the category of representations. Discuss the classification problem, in particular explaining what is wild about wild quivers (and mention the classification of representations for (extended) Dynkin quivers in passing). Literature: e.g. [CB1], [K], [Re], [Sch].
Define Kac polynomials. In particular, explain the arithmetic of representations over finite fields and sketch the existence proof for Kac polynomials. Give some simple examples. Formulate the Kac conjecture and explain their relevance w.r.t. the classification problem. Formulate which cases of these conjectures were known before [HRV]. Literature: [CBVdB], [Hu], [K1], [K2], [K3], [HRV], [Mo], [Re].
Introduce the geometric point of view on representations. Recall the main methods of Geometric Invariant Theory (and symplectic reduction) which are needed. Literature: [CB2], [CB3], [KR], [Mu], [Re1].
Define Nakajima quiver varieties and mention why Nakajima studied them.Some basic results on preprojective varieties have to be mentioned, too. Give some examples. Literature: [CB3], [CBH], [Na].
Define the Weyl group action on quiver varieties, because that's crucial for [HRV], following Nakajima, Lusztig, Maffei, CrawleyBoeveyHolland. Literature: [CBH], [Na1], [Lu], [Ma],
Define Schur polynomials and HallLittlewood polynomials using orthogonality in the ring of symmetric functions. Discuss how this yields the character theory of S_n and GL_n(F_q). Literature: [M].
State the main result of [HRV] in detail, introducing the relevant quiver varieties and symmetric functions. Compare with the main result of [CBVdB]. Give some examples. Literature: [CBVdB], [HRV].
Discuss the methods of [CBVdB], in
particular with respect to purity of quiver varieties and the
polynomial count property. Literature: [CBVdB].
Discuss the previous work of Hausel, Letellier and RodriguezVillegas, in particular the proof of the first Kac conjecture and Fourier transform over finite fields. Literature: [H], [HRV1], [HRV2], [L].
To work through the proof of Theorem 2.3, introduce all "standard" facts on ladic cohomology (etale sheaves and their cohomology, Grothendieck trace formula for Frobenius), as well as the necessary hyperkaehler methods. Literature: t.b.a.
Sketch the rest of the proof. More details later.
[CB1] CrawleyBoevey, William. Lectures on representations of quivers. http://www1.maths.leeds.ac.uk/~pmtwc/
[CB2] CrawleyBoevey, William. Geometry of representations of algebras. http://www1.maths.leeds.ac.uk/~pmtwc/
[CB3] CrawleyBoevey, William. Geometry of the moment map for representations of quivers. Compositio Math. 126 (2001), no. 3, 257293.
[CBH] CrawleyBoevey, William; Holland, Martin. Noncommutative deformations of Kleinian singularities. Duke Math. J. 92 (1998), no. 3, 605635.
[CBVdB] CrawleyBoevey, William; Van den Bergh, Michel. Absolutely indecomposable representations and KacMoody Lie algebras. With an appendix by Hiraku Nakajima. Invent. Math. 155 (2004), no. 3, 537559.
[D] Davison, Ben. Purity of critical cohomology and Kac's conjecture. Preprint 2013. arXiv:1311.6989
[H] Hausel, Tamás. Kac conjecture from Nakajima quiver varieties. Inv. Math. 181 (2010), no. 1, 2137.
[HRV] Hausel, Tamás; Letellier, Emmanuel; RodriguezVillegas, Fernando. Positivity for Kac polynomials and DTinvariants of quivers. Ann. of Math. (2) 177 (2013), no. 3, 1147—1168.
[HRV1] Hausel, Tamás; Letellier, Emmanuel; RodriguezVillegas, Fernando. Arithmetic harmonic analysis on character and quiver varieties II. Adv. Math. 234 (2013), 85—128.
[HRV2] Hausel, Tamás; Letellier, Emmanuel; RodriguezVillegas, Fernando. Arithmetic harmonic analysis on character and quiver varieties. Duke Math. J. 160 (2011), no. 2, 323—400.
[HRV3] Hausel, Tamás; Letellier, Emmanuel; Rodriguez Villegas, Fernando. Topology of character varieties and representation of quivers. C. R. Math. Acad. Sci. Paris 348 (2010), no. 34, 131—135.
[Hu] Hua, Jiuzhao. Counting representations of quivers over finite fields. J. Algebra 226 (2000), 10111033.
[K1] Kac, V. G. Infinite root systems, representations of graphs and invariant theory. J. Algebra 78 (1982), no. 1, 141162.
[K2] Kac, V. G. Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56 (1980), no. 1, 57—92.
[K3] Kac, V. G. Root systems, representations of quivers and invariant theory. Invariant theory (Montecatini, 1982), 74108, Lecture Notes in Mathematics, 996, SpringerVerlag, 1983.
[K] Kraft, Hanspeter Geometrische Methoden in der Invariantentheorie. Vieweg, 1984.
[KR] Kraft, Hanspeter; Riedtmann, Christine: Geometry of representations of quivers. Representations of algebras (Durham, 1985), 109145, London Math. Soc. Lecture Note Ser., 116, Cambridge Univ. Press, Cambridge, 1986.
[Le] Letellier, Emmanuel. Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras. Lecture Notes in Mathematics, 1859, SpringerVerlag, 2005.
[Lu] Lusztig, George. Quiver varieties and Weyl group actions. Ann. Inst. Fourier (Grenoble) 50 (2000), 461489.
[M] Macdonald, Ian G. Symmetric functions and Hall polynomials. Second ed., The Clarendon Press, Oxford University Press, New York 1995.
[Ma] Maffei, Andrea. A remark on quiver varieties and Weyl groups. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 3, 649686.
[Mo] Mozgovoy, Sergey. Motivic DonaldsonThomas invariants and the Kac conjecture. Compos. Math. 149 (2013), no. 3, 495504.
[Mu] Mukai, Shigeru. An introduction to invariants and moduli. Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. Cambridge Studies in Advanced Mathematics, 81. Cambridge University Press, Cambridge, 2003.
[Na] Nakajima, Hiraku. Instantons on ALE spaces, quiver varieties, and KacMoody algebras. Duke Math. J. 76 (1994), no. 2, 365416.
[Na1] Nakajima, Hiraku. Reflection functors for quiver varieties and Weyl group actions. Math. Ann. 327 (2003), no. 4, 671721.
[Re] Reineke, Markus: The use of geometric and quantum group techniques for wild quivers. Representations of finite dimensional algebras and related topics in Lie theory and geometry, 365–390, Fields Inst. Commun., 40, Amer. Math. Soc., Providence, RI, 2004.
[Re1] Reineke, Markus: Moduli of representations of quivers. Trends in representation theory of algebras and related topics, 589637, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008.
[Sch] Ralf Schiffler. Quiver representations (CMS books in mathematics), Springer, 2014.

Monday 
Tuesday 
Wednesday 
Thursday 
Friday 
09:00 – 10:30 

V. Makam 
M. Maslovaric 
G. Zhao 
N.N. 
10:30 – 10:45 
Coffee 
Coffee 
Coffee 
Coffee 

10:45 – 12:15 
Ö. Eiriksson 
A. Dönmez 
H. Franzen 
S. Mozgovoy 

12:30 – 14:00 
Lunch 
Lunch 
Lunch 
Lunch 
Lunch 
14:00 – 15:30 
H. Vogel 
E. Murphy 

A. Minets 

15:30 – 16:00 
Coffee 
Coffee 
Coffee 

16:00 – 17:30 
N.N. 
G. Amazeen 
S. Mozgovoy 

18:00 – 19:00 
Dinner 
Dinner 
Dinner 
Dinner 
Name 
First name 
University 
Amazeen 
Gretar 
Berlin 
Boos 
Lena 
Wuppertal 
Dönmez 
Arif 
Wuppertal 
Eiriksson 
Ögmunudur 
Bielefeld 
Franzen 
Hans 
Bonn 
König 
Steffen 
Stuttgart 
Makam 
Viswambhara 
Michigan 
Maslovaric 
Marcel 
Göttingen 
Minets 
Alexander 
Paris 
Mozgovoy 
Sergey 
Dublin 
Murphy 
Eoin 
Sheffield 
Reineke 
Markus 
Wuppertal 
Rompf 
Daniel 
Wuppertal 
Sabonis 
Deividas 
München 
Vogel 
Hannah 
Graz 
Zhao 
Gufang 
Paris 
The funding organizations plan to cover accomodation and full boarding. However, travel expenses usually can not be covered. If you want to participate in the Spring School, please send an email to M. Reineke (mreineke “at” uniwuppertal.de), if possible indicating which topic you are willing to give a talk about.
The deadline for registration was 15 January 2015.
M. Reineke, 12 February 2015