Spring School :

“Kac conjectures and quiver varieties”

CVJM Bildungsstätte Bundeshöhe,

Wuppertal, 23 - 27 March 2015

Organizers: Steffen König (Stuttgart), Markus Reineke (Wuppertal)

Hosted and funded by:

Research group Algebra/Number Theory, BU Wuppertal,

SPP 1388 “Representation Theory”




The Spring School:

The aim of the Spring School is to work through the proof, by T. Hausel, E. Letellier and F. Rodriguez-Villegas, 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.


Programme:

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.

Topic 1: Quivers and their representations (H. Vogel)

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].

Topic 2: Kac polynomials (N.N.)

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].

Topic 3: Geometry of representations (V. Makam)

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].

Topic 4: Nakajima quiver varieties (Ö. Eiriksson)

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].

Topic 5: Weyl group action on quiver varieties (E. Murphy)

Define the Weyl group action on quiver varieties, because that's crucial for [HRV], following Nakajima, Lusztig, Maffei, Crawley-Boevey-Holland. Literature: [CBH], [Na1], [Lu], [Ma],

Topic 6: Hall-Littlewood polynomials and characters of symmetric and general linear groups (G. Amazeen)

Define Schur polynomials and Hall-Littlewood 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].

Topic 7: Formulation of the results of Hausel, Letellier and Rodriguez-Villegas (M. Maslovaric)

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].

Topic 8: The Kac conjectures for indivisible dimension vectors (A. Dönmez)

Discuss the methods of [CBVdB], in particular with respect to purity of quiver varieties and the
polynomial count property. Literature: [CBVdB].

Topic 9: Proof of the first Kac conjecture and arithmetic Fourier transform (G. Zhao)

Discuss the previous work of Hausel, Letellier and Rodriguez-Villegas, in particular the proof of the first Kac conjecture and Fourier transform over finite fields. Literature: [H], [HRV1], [HRV2], [L].

Topic 10: Proof of the second Kac conjecture pt. 1 (H. Franzen, A. Minets)

To work through the proof of Theorem 2.3, introduce all "standard" facts on l-adic cohomology (etale sheaves and their cohomology, Grothendieck trace formula for Frobenius), as well as the necessary hyperkaehler methods. Literature: t.b.a.

Topic 11: Proof of the second Kac conjecture pt. 2 (H. Franzen, A. Minets)

Sketch the rest of the proof. More details later.


References:

                




Schedule:


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




Participants:

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” uni-wuppertal.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