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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
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 padic 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, Ktheory 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 Dmodules from the microlocal point of view. His research on real groups, joint with Schmid, includes the proof of the BarbaschVogan 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 KnizhnikZamolodchikov equations, Cohomological Hall algebras, and vertex algebras associated to toric CalabiYau 3folds.
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 KhovanovLaudaRouquier Algebras III: Symmetric Affine Type. arXiv. Math. Z. 287 (2017), no. 12, 243–286.
 Peter J. McNamara, Finite Dimensional Representations of KhovanovLaudaRouquier 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), 257283.
 A. Kleshchev, A. Mathas, and Arun Ram, Universal Specht modules for cyclotomic Hecke algebras , arXiv1102.3519, Proc. London Math. Soc. (3) 105 (2012) 12451289.
 P. Diaconis and Arun Ram, A probabilistic interpretation of the Macdonald polynomials, arXiv1007.4779, The Annals of Probability 40 (2012) Vol. 40 No. 5, 18611896.
 Roman Bezrukavnikov and Kari Vilonen, Koszul Duality for Quasisplit Real Groups, arXiv:1510.08343, Under revision for Invent. Math.
 Masaki Kashiwara and Kari Vilonen, Microdifferential systems and the codimensionthree conjecture. Ann. of Math. (2) 180 (2014) no. 2, 573620.
 Wilfried Schmid and Kari Vilonen, Hodge theory and unitary representations of reductive Lie groups. Frontiers of mathematical sciences, 397420, Int. Press, Somerville, MA, 2011, arXiv
 Kari Vilonen and Ting Xue, Character sheaves for symmetric pairs, arXiv:1806.02506
 TsaoHsien Chen, Kari Vilonen and Ting Xue, Springer correspondence for the split symmetric pair in type A , Compos. Math. 154 (2018), no. 11, 24032425. arXiv
 TsaoHsien 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), 515533. 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, 10291074.

Representation theory seminar 2020, Semester 2
Basic Notions in Representation Theory
Organisers: Kari Vilonen, Ting Xue
Topics: We will cover basic notions in representation theory going back to the geometric methods introduced around 1980. We will be discussing BeilinsonBernstein localization, the KazhdanLusztig conjectures, Koszul duality, and the Jantzen conjectures. The goal of the seminar is for people to learn and think about the topics we discuss. An original reference, prior to the introduction of Koszul duality, is Beilinson’s ICM talk in 1983. Here is the original BeilinsonGinzburg preprint on Koszul duality.
Time: Thursdays 2:15pm4:15pm.
Location: Online
Dec 17 Anna Romanov (Sydney) JordanHolder multiplicities of Verma modules with rational highest weight: continued Notes
Dougal Davis (Edinburgh) Recap: category O to monodromic Dmodules to mixed Hodge modules to the Hecke algebra Notes
Dec 10 No seminar (AustMS meeting)
Dec 3 Anna Romanov (Sydney) JordanHolder multiplicities of Verma modules with rational highest weight Notes
Abstract: Earlier this semester, Gufang explained how we can compute the Jordan—Holder multiplicities of Verma modules in the principal block of category O by realising them in terms of constructible sheaves on the flag variety. In this talk, I will attempt to do the same thing for Verma modules with regular rational highest weight. Instead of sheaves on the flag variety, our topological players will be sheaves on line bundles over the flag variety with some extra structure. I will start by explaining the strategy laid out in Lusztig’s chapter “Computation of local intersection cohomology of certain line bundles over a Schubert variety”, then I will sketch another approach using only Dmodules.
November 26 Dougal Davis (Edinburgh) Mixed geometry over finite fields: continued Notes
Kari Vilonen (Melbourne) Beilinson functors Notes
November 19 Kari Vilonen (Melbourne) Punctural purity and the socle filtration Notes
Dougal Davis (Edinburgh) Mixed geometry over finite fields Notes
November 12 Kari Vilonen (Melbourne) The Jantzen conjecture: continued Notes
November 5 Kari Vilonen (Melbourne) The Jantzen conjecture: continued Notes
October 29 Kari Vilonen (Melbourne) Comments on mixed geometry and the Jantzen conjecture Notes
Abstract: I will first make a few comments related to Gufang’s talk last week. I will recall some material from the previous talks and will discuss and formulate the Jantzen conjecture. I will also make some preliminary remarks to prepare for its proof.
October 22 Gufang Zhao (Melbourne) Koszulity of category O via mixed geometry: continued Notes
October 15 No seminar
October 8 Gufang Zhao (Melbourne) Koszulity of category O via mixed geometry Notes (preseminar) Notes (inseminar)
Abstract: In this talk we review some basic facts about mixed Hodge modules. After that, we follow Section 4 of Beilinson, Ginzburg, Soergel’s paper to construct a graded lift of the principal block of category O using mixed Hodge modules, and then prove the Koszul property of the graded lift.
October 1 Kari Vilonen (Melbourne) Interlude Notes
September 24 Ting Xue (Melbourne) BGG category and Koszul duality: continued Notes
September 17 Dougal Davis (Edinburgh) BeilinsonBernstein localisation again: continued Notes
Ting Xue (Melbourne) BGG category and Koszul duality Notes
Abstract: The study of BGG category , introduced in the 1970s by Joseph Bernstein, Israel Gelfand, and Sergei Gelfand, “offers a rewarding tour of the beautiful terrain that lies just beyond the classical CartanWeyl theory of finite dimensional representations of ” (Humphreys). In the first part of the talk we will review the basic properties of the BGG category , duality in , BGG reciprocity, blocks of , and translation functors etc. In the second part we will discuss the Koszul selfduality of the principal block of category . We will also try to make connections to the previous lectures when we can.
September 10 Yaping Yang (Melbourne) An introduction to the Riemann–Hilbert correspondence: continued Notes
Dougal Davis (Edinburgh) BeilinsonBernstein localisation again: continued Notes
September 3 (2pm4pm) Note early start time!
Yaping Yang (Melbourne) An introduction to the Riemann–Hilbert correspondence: continued Notes
Dougal Davis (Edinburgh) BeilinsonBernstein localisation again Notes
Abstract: In this talk, I will go back over BeilinsonBernstein localisation in a bit more detail. This time, I will cover the general definition of twisted differential operators for nonintegral weights, and the closely related notion of monodromic Dmodules. With the aid of some very explicit examples for , I will also explain how to go back and forth between representations of the Lie algebra and monodromic Dmodules on the flag variety in practice.
August 27 Gufang Zhao (Melbourne) Proof of KazhdanLusztig conjectures on the Hecke algebra: continued
Yaping Yang (Melbourne) An introduction to the Riemann–Hilbert correspondence Notes
Abstract: I will give a general introduction to the Riemann–Hilbert correspondence. Let be an algebraic variety over and let be the corresponding complex analytic variety in classical topology. The Riemann–Hilbert correspondence for regular singular connections was proved by Deligne in 1970. It is an equivalence between the category of flat connections on algebraic vector bundles on with regular singularities and the category of local systems on . More generally, the Riemann–Hilbert correspondence for regular holonomic Dmodules was proved by Kashiwara and Mebkhout independently in 1984. It says the de Rham functor induces an equivalence from the category of regular holonomic Dmodules on to the category of perverse sheaves on .
August 20 Peter McNamara (Melbourne) BeilinsonBernstein localisation: continued Notes
Gufang Zhao (Melbourne) Proof of KazhdanLusztig conjectures on the Hecke algebra Notes
Abstract: In this talk, we recall KazhdanLusztig conjectures on the Hecke algebra. Using the BeilinsonBernstein localization theorem from Peter’s talk, and the RiemannHilbert correspondence, we give a description of the irreducible modules and Verma modules of the Lie algebra in terms of constructible sheaves on the flag variety. Then, we describe an assignment of a Hecke algebra element to each complex of constructible sheaf. We use the decomposition theorem to show some properties of this assignment with respect to IC sheaves and constant sheaves of Schubert varieties. Finally we deduce KazhdanLusztig conjecture expressing the coefficients of irreducible modules with respect to Verma modules of a Lie algebra in terms of Hecke algebras, and show the nonnegativity of the coefficients.
August 13 Peter McNamara (Melbourne) BeilinsonBernstein localisation Notes
Abstract: I will talk about the Beilinson Bernstein localisation theorem which gives an equivalence of categories between a category of representations of a Lie algebra, and a category of Dmodules on the flag variety (we will incorporate the twist into the story). This theorem has since been a fundamental result in geometric representation theory. Examples will be provided when possible.
Past Seminars:
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 VershikOkounkov approach.
Time: Wednesdays 121pm
Location: Zoom (Please contact the organisers for the link)
Schedule
10 Feb Zhongtian Chen Universal positive selfadjoint Hopf algebra: uniqueness.
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
Date  Topics  References  Speaker 
Oct 14/21/28  Representations of finite groups, characters, symmetric groups  [Za] A.1, A.1.2  Yifan Guo 
Dec 2  Frobenius’s formula and applications  [Za] A.1.3  Davood Nejaty 
Dec 9/16  Hopf algebras, irreducible, and primitive elements  [Ze] 1.3 2. (Definition 1.4, Theorem 2.2 and briefly its proof)  Simon Thomas 
Feb 10, 2021  Universal positive selfadjoint Hopf algebra: uniqueness.  [Ze] 3  Zhongtian Chen 
Universal positive selfadjoint Hopf algebra: special elements.  [Ze] 4  Ennes Mehmedbasic  
Symmetric functions, induction and restriction.  [Ze] 5.3, 6  Kshitija Vaidya  
GelfandZeitlin algebra and YoungJucysMurphy elements.  [OV] 1, 2.  Weiying Guo  
Degenerate affine Hecke algebra and representations.  [OV] 3, 4.  Adam Monteleone  
Branching theorem.  [OV] 5, 6.  Davood Nejaty  
Branching rule and Murnaghan–Nakayama rule.  [OV] 7, 8  tba  
References  [VO] A. M. Vershik and A. Yu. Okounkov, A New Approach to the Representation Theory of the Symmetric Groups. II, arXiv:math.RT/0503040.
[Za] D. Zagier, Applications of the representation theory of finite groups, appendix to Graphs on Surfaces and Their Applications, (2004). [Ze] A. Zelevinsky, Representations of Finite Classical Groups: A Hopf Algebra Approach, (1981).

Dougal Davis (University of Edinburgh), Mar. 30, 2020Apr. 3, 2020
Xinwen Zhu (Caltech), November 24Dec 1, 2019
ChengChiang Tsai (Stanford University), Nov 19Dec 8, 2019
Geordie Williamson (Sydney), October 2628, 2019
Bill Casselman (University of British Columbia), Oct 2427, 2019
Emily Norton (MPIM), February 19March 2, 2019
Luca Migliorini (University of Bologna), December 39, 2018
ChengChiang Tsai (Stanford University), Nov 3Dec 1, 2018
Xinwen Zhu (Caltech), August 2330, 2018
Anthony Henderson (Sydney), August 2124, 2018
Jessica Fintzen (University of Michigan), November 27December 3, 2017
Sam Raskin (University of Texas at Austin) , September 1222, 2017
Takuro Mochizuki (RIMS, Kyoto University), September 415, 2017
Carl Mautner (UC Riverside), August 1626, 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 coorganising (with Masoud Kamgarpour and Peng Shan) the MATRIX workshop Frontiers in Representation Theory, 1425 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 CalabiYau 3folds” at the GRT at Home seminar on 23 June, 2020.
 Nora Ganter, Yaping Yang, and Gufang Zhao coorganized with Daniel Berwick Evans and Theo JohnsonFreyd the workshop on elliptic cohomology and physics 2528 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 1620 2019, Yaping Yang and Gufang Zhao coorganised 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 FebMar 2019. During the week of February 25March 1, 2019, Yaping Yang coorganised 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.