Members

 Peter McNamara                          Arun Ram                                        Kari Vilonen

Ting Xue                                           Yaping Yang                                     Gufang Zhao

Affiliated Members

Jan de Gier        Nora Ganter        Alex Ghitza      Christian Haesemeyer     Jack Hall

 Thomas Quella        David Ridout           Chenyan Wu        Paul Zinn-Justin

Peter McNamara‘s research is in categorical and geometric representation theory. He has worked on a range of topics including quantum groups and their categorifications, perverse sheaves and Schubert varieties, algebraic combinatorics and p-adic groups.

Arun Ram’s research is in the area of Combinatorial Representation Theory. Topics he has worked on include tableaux combinatorics, crystals, diagram algebras, Lie theory, quantum groups, Hecke algebras, Schubert calculus, K-theory and cohomology of flag varieties and affine flag varieties.  Current projects include constructions of representations of quantum affine algebras and a study of the combinatorics of double affine Hecke algebras and Macdonald polynomials.

Kari Vilonen‘s research is in the areas of real groups, the Langlands program, and related algebraic geometry. He has worked on several aspects of the geometric Langlands program and on more foundational questions on perverse sheaves and D-modules from the microlocal point of view. His research on real groups, joint with Schmid, includes the proof of the Barbasch-Vogan conjecture and a conjectural theory of Hodge structures on representations of real groups.

Ting Xue‘s research is in the areas of representation theory and algebraic groups. She has worked on questions related to geometry of nilpotent orbits  and Springer theory, including small or bad characteristics. She is also interested in combinatorics arising from representation theory.

Yaping Yang‘s research lies in the area of Lie algebras, geometric representation theory, quantum groups, and the related geometry and topology. Her current work includes Knizhnik-Zamolodchikov equations, Cohomological Hall algebras, and vertex algebras associated to toric Calabi-Yau 3-folds.

Gufang Zhao‘s research lies at the interface between algebraic geometry and representation theory. More specifically, he has been working on projects concerning derived category of coherent sheaves, oriented cohomology theories of algebraic varieties, and their applications in representation theory. He is also fond of varieties of local systems and instantons, quantum integrable systems, and related aspects in mathematical physics.

Selected papers

• A full list of each member’s publications can be found on their individual webpages.
  • • M. Lanini and P. J. McNamara, Singularities of Schubert varieties within a right cell. arXiv2003.08616.
    • P. J. McNamara, Representation Theory of Geometric Extension Algebras, arXiv:1701.07949.
    • P. J. McNamara. Representations of Khovanov-Lauda-Rouquier Algebras III: Symmetric Affine Type. arXiv. Math. Z. 287 (2017), no. 1-2, 243–286.
    • Peter J. McNamara, Finite Dimensional Representations of Khovanov-Lauda-Rouquier Algebras I: Finite Type, arXiv. J. Reine Angew. Math. 707 (2015), 103–124.
    • D. George, Arun Ram, J. Thompson and R. Volkas, Symmetry breaking, subgroup embeddings and the Weyl group, arXiv1203.1048, Physical Review D 87 105009 (2013) [14 pages]
    • Z. Daugherty, Arun Ram and R. Virk, Affine and degenerate affine BMW algebras: The center, arXiv1105.4207, Osaka J. Math 51 (2014), 257-283.
    • A. Kleshchev, A. Mathas, and Arun Ram, Universal Specht modules for cyclotomic Hecke algebras , arXiv1102.3519, Proc. London Math. Soc. (3) 105 (2012) 1245-1289.
    • P. Diaconis and Arun Ram, A probabilistic interpretation of the Macdonald polynomials, arXiv1007.4779, The Annals of Probability 40 (2012) Vol. 40 No. 5, 1861-1896.
    • Roman Bezrukavnikov and Kari Vilonen, Koszul Duality for Quasi-split Real Groups, arXiv:1510.08343, Under revision for Invent. Math.
    • Masaki Kashiwara and Kari Vilonen, Microdifferential systems and the codimension-three conjecture. Ann. of Math. (2) 180 (2014) no. 2, 573-620.
    • Wilfried Schmid and Kari Vilonen, Hodge theory and unitary representations of reductive Lie groups. Frontiers of mathematical sciences, 397-420, Int. Press, Somerville, MA, 2011, arXiv
    • Kari Vilonen and Ting Xue, Character sheaves for symmetric pairs, arXiv:1806.02506
    • Tsao-Hsien Chen, Kari Vilonen and Ting Xue, Springer correspondence for the split symmetric pair in type A , Compos. Math. 154 (2018), no. 11, 2403-2425. arXiv
    • Tsao-Hsien Chen, Kari Vilonen and Ting Xue, On the cohomology of Fano varieties and the Springer correspondence, With an appendix by Dennis Stanton. Adv. Math. 318 (2017), 515-533. arXiv.
    • Ting Xue, Combinatorics of the Springer correspondence for classical Lie algebras and their duals in characteristic 2. Adv. Math. 230 (2012) no. 1, 229–262.
    • Marc Levine, Yaping Yang, and Gufang Zhao, Algebraic Elliptic cohomology theory and flops 1, appendix by Joël Riou. Mathematische Annalen volume 375, pages 1823–1855 (2019).
    • Miroslav Rapcak, Yan Soibelman, Yaping Yang, and Gufang Zhao, Cohomological Hall algebras, vertex algebras and instantons. Communications in Mathematical Physics volume 376, pages 1803–1873 (2020)
    • Yaping Yang, and Gufang Zhao, The cohomological Hall algebras for a preprojective algebra. Proc. Lond. Math. Soc. 116, 1029-1074.

Representation theory seminar 2021, Semester 1
Basic Notions in Representation Theory

Organisers: Kari Vilonen, Ting Xue, Yaping Yang

Topics: Affine Lie algebras, Kazhdan-Lusztig conjectures, localisation

Time: Tuesdays 3:15pm-5pm.

8 June 2:15pm Arun Ram (Melbourne) Representations of affine Hecke algebras III: Standard modules 

Abstract: Most standard modules are built by induction.  I’ll describe principal series modules and the basic theorems about their simplicity and their composition factors.  Then I’ll define tempered  and square integrable modules, and explain how the standard modules correspond to generalized Springer fibers and the square integrable modules correspond to cuspidal nilpotent elements. This provides an indexing of irreducible representations of affine Hecke algebras by Deligne-Langlands parameters.

1 June 2:15pm (note time change!) Arun Ram (Melbourne) Representations of affine Hecke algebras II Central characters, weight spaces and intertwiners Notes

Abstract:  I will try to explain how the representation theory of the affine Hecke algebra is coded by local regions in a hyperplane arrangement, sometimes called the Shi arrangement. I’ll introduce the favourite induced modules (standard modules) and explain how to study their structure by weight spaces and intertwiners.

25 May Arun Ram (Melbourne) Representations of affine Hecke algebras I Notes

Abstract: In this first talk, I’ll define the isogeneous affine Hecke algebras and explain how to compare their representations. Then I’ll explain how to get the representations of finite Hecke algebras of the finite complex reflection groups G(r,p,n) from the affine Hecke algebra of type GL_n.

18 May Anna Romanov (Sydney) The infinite-dimensional geometric story: Kac-Moody groups, affine flag varieties, and D-modules, continued Notes

11 May Arun Ram (Melbourne) Decomposition numbers for standard objects in categories O Notes

Abstract: The Verma modules are indexed by their highest weight. They have a simple quotient and so the irreducibles are indexed by their
highest weight.  The composition factors of a Verma module must all lie in the same orbit of the Weyl group.  In the affine case the orbits take three different shapes depending on whether it is positive level, negative level, or critical level.  In each case there is a different family of Kazhdan-Lusztig type polynomials that describes the multiplicity of the irreducible in the layers of the Jantzen filtration of the Verma module.  I’ll try to explain what these affine Weyl group orbits and Kazhdan-Lusztig polynomials are.

4 May Anna Romanov (Sydney) The infinite-dimensional geometric story: Kac-Moody groups, affine flag varieties, and D-modules, continued Notes

27 April Anna Romanov (Sydney) The infinite-dimensional geometric story: Kac-Moody groups, affine flag varieties, and D-modules Notes

Abstract: In these talks, I will describe some geometric objects that play a role in the representation theory of Kac-Moody Lie algebras. In the Kac-Moody setting, we have no Beilinson—Bernstein localisation theorem. However, as Yaping told us last week, D-modules on certain infinite-dimensional varieties can still be used to prove results about representations of affine Lie algebras.  This series of talks has two goals: (1) to clarify the state of affairs on Beilinson—Bernstein-type equivalences of categories for affine Lie algebras, and (2) sketch the proof (due to Kashiwara-Tanisaki) that negative level blocks of category O for an affine Lie algebra are equivalent to certain categories of equivariant D-modules on the affine flag variety.

In the first week, I will define Kac-Moody groups and their flag varieties carefully, describe their ind-variety structure, and do an SL2 example.

In the second week, I will introduce D-modules, state the known results about equivalences of categories between D-modules and representations of affine Lie algebras, and sketch the Kashiwara-Tanisaki negative-level proof.

20 April Yaping Yang (Melbourne)  Kazhdan-Lusztig Conjecture for Kac-Moody Lie algebras: continued Notes

13 April Yaping Yang (Melbourne)  Kazhdan-Lusztig Conjecture for Kac-Moody Lie algebras

Abstract:  In my talk, I will review the Kac Moody Lie algebras, the affine root systems, and the affine Weyl groups. I will state the Kazhdan-Lusztig Conjecture for Kac-Moody Lie algebras. In particular, at the non-critical level, the transition matrix of the characters of the irreducible highest weight modules and the characters of the Verma modules is given by the affine Kazhdan-Lusztig polynomial.

30 Mar Kari Vilonen (Melbourne) Summary and conclusion of the Jantzen conjecture

23 Mar Kari Vilonen (Melbourne) the Jantzen conjecture

Past Seminars:

2020 Semester 2

Representation Theory Student Seminar 2021 Semester 1

Organisers: Ting Xue, Gufang Zhao

Overview

This is a learning seminar on representation theory, aimed at 3rd year undergraduate students, MSc students, as well as PhD students in mathematics or related fields.

In the learning seminar, participants are expected to learn a topic based on the references given, and present their work to the other participants. The audience are expected to ask questions and make comments during the presentation. Discussions tangential to the topics are welcome. The idea is for all participants to actively learn and discuss together.

This semester, the topic is representation theory of symmetric groups. We assume basic knowledge of group theory and linear algebra. The seminar will begin with basic notions of representations of symmetric groups and symmetric functions, and statements of theorems on characters. Then, we will discuss two different approaches to establish these theorems, Zelevinsky’s approach using Hopf algebras and the Vershik-Okounkov approach.

Time: Wednesdays 13:15-14:15pm

Location: Peter Hall 213 and/or Zoom (Please contact the organisers for the link)

Schedule

2 June Peter McNamara Overview

26 May Davood Nejaty Branching theorem, continued Notes

19 May Adam Monteleone Degenerate affine Hecke algebra and representations, continued Notes

Davood Nejaty Branching theorem, continued

12 May Davood Nejaty Branching theorem Notes

5 May Adam Monteleone Degenerate affine Hecke algebra and representations

28 Apr Weiying Guo Gelfand-Tsetlin algebra and Young-Jucys-Murphy elements, continued. Notes

21 Apr Weiying Guo Gelfand-Tsetlin algebra and Young-Jucys-Murphy elements, continued.

14 Apr Weiying Guo Gelfand-Tsetlin algebra and Young-Jucys-Murphy elements Notes

7 Apr Ennes Mehmedbasic PSH algebra and combinatorial rules

31 Mar Kshitija Vaidya Symmetric functions, continued. Notes

17 Mar Kshitija Vaidya Symmetric functions, continued. Notes

10 Mar Kshitija Vaidya Symmetric functions Notes

3 Mar  Zhongtian Chen Universal positive self-adjoint Hopf algebra: uniqueness, continued. Notes

24 Feb Zhongtian Chen Universal positive self-adjoint Hopf algebra: uniqueness, continued. Notes

17 Feb Zhongtian Chen Universal positive self-adjoint Hopf algebra: uniqueness, continued. Notes

10 Feb Zhongtian Chen Universal positive self-adjoint Hopf algebra: uniqueness. Notes

Representation Theory Student Seminar 2020 Semester 2

16 Dec Simon Thomas Hopf algebras, irreducible, and primitive elements, II Notes

9 Dec Simon Thomas Hopf algebras, irreducible, and primitive elements, I Notes

2 Dec Davood Nejaty Frobenius’s formula and applications to topology Notes

28 Oct Yifan Guo Representations of finite groups, characters, symmetric groups, III Notes

21 Oct Yifan Guo Representations of finite groups, characters, symmetric groups, II Notes

14 Oct Yifan Guo Representations of finite groups, characters, symmetric groups, I Notes

30 Sep Organisational meeting

Dougal Davis (University of Edinburgh), Mar. 30, 2020-Apr. 3, 2020

Xinwen Zhu (Caltech), November 24-Dec 1, 2019

Cheng-Chiang Tsai (Stanford University), Nov 19-Dec 8, 2019

Geordie Williamson (Sydney), October 26-28, 2019

Bill Casselman (University of British Columbia), Oct 24-27, 2019

Emily Norton (MPIM), February 19-March 2, 2019

Luca Migliorini (University of Bologna), December 3-9, 2018

Cheng-Chiang Tsai (Stanford University), Nov 3-Dec 1, 2018

Xinwen Zhu (Caltech), August 23-30, 2018

Anthony Henderson (Sydney), August 21-24, 2018

Jessica Fintzen (University of Michigan),  November 27-December 3, 2017

Sam Raskin (University of Texas at Austin) , September 12-22, 2017

Takuro Mochizuki (RIMS, Kyoto University), September 4-15, 2017

Carl Mautner (UC Riverside), August 16-26, 2017

• Kari Vilonen was awarded a highly prestigious ARC Laureate Fellowship on Real groups and the Langlands program.
• Peter McNamara gave a talk on June 25, 2020 at the informal Friday seminar at U Sydney.
• Nora Ganter, Peter McNamara, Yaping Yang and Gufang Zhao are co-organising (with Masoud Kamgarpour and Peng Shan) the MATRIX workshop Frontiers in Representation Theory, 14-25 February 2022.
• Ting Xue will be a speaker at AMSI Winter School 2020, New directions in representation theory, University of Queensland. (Postponed.)
• Yaping Yang gave a talk titled “Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds” at the GRT at Home seminar on 23 June, 2020.
• Nora Ganter, Yaping Yang, and Gufang Zhao co-organized with Daniel Berwick Evans and Theo Johnson-Freyd the workshop on elliptic cohomology and physics 25-28 May 2020.
• Arun Ram gave a performance of “Mendelssohn Salon 1828” with pianist Michael Leslie on
  12 March 2020 at Tempo Rubato.
• From Dec 2019 to Feb.29 2020, Yaping Yang and Gufang Zhao visited the Kavli Institute for the Physics and Mathematics of the Universe (IPMU), Japan. During the visit, Gufang Zhao gave a seminar talk titled “Cohomological Hall algebras and their representation theories” at the Mathematics and String Theory Seminar at IPMU.
• The week of December 16-20 2019, Yaping Yang and Gufang Zhao co-organised the workshop on “Geometric Representation Theory and Quantum Field Theory” together with Hiraku Nakajima (IPMU), Peng Shan (Tsinghua), Wenbin Yan (Tsinghua) at TSIMF, Sanya, China.
• Yaping Yang and Gufang Zhao visited the Perimeter Institute for Theoretical Physics, Waterloo, Canada during Feb-Mar 2019. During the week of February 25-March 1, 2019, Yaping Yang co-organised the workshop “Cohomological Hall algebras in Mathematics and Physics” at Perimeter Institute (with Kevin Costello (PI) and Yan Soibelman (KSU)).
• Yaping Yang received an ARC Discovery Early Career Award (DE 190101231) in Dec 2018.
• Gufang Zhao received an ARC Discovery Early Career Award (DE 190101222) in Dec 2018.