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Members

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

- - 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 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 Beilinson-Bernstein localization, the Kazhdan-Lusztig 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 Beilinson-Ginzburg preprint on Koszul duality.

Time: Thursdays 2:15pm-4:15pm.

Location: Online

Dec 17 Anna Romanov (Sydney) Jordan-Holder multiplicities of Verma modules with rational highest weight: continued Notes

Dougal Davis (Edinburgh) Re-cap: category O to monodromic D-modules to mixed Hodge modules to the Hecke algebra Notes

Dec 10 No seminar (AustMS meeting)

Dec 3 Anna Romanov (Sydney) Jordan-Holder 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 D-modules.

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 (pre-seminar) Notes (in-seminar)

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 $\mathcal{O}$ and Koszul duality: continued Notes

September 17 Dougal Davis (Edinburgh) Beilinson-Bernstein localisation again: continued Notes

Ting Xue (Melbourne) BGG category $\mathcal{O}$ and Koszul duality Notes

Abstract: The study of BGG category $\mathcal{O}$ , 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 Cartan-Weyl theory of finite dimensional representations of $\mathfrak{g}$ ” (Humphreys). In the first part of the talk we will review the basic properties of the BGG category $\mathcal{O}$ , duality in $\mathcal{O}$ , BGG reciprocity, blocks of $\mathcal{O}$ , and translation functors etc. In the second part we will discuss the Koszul self-duality of the principal block of category $\mathcal{O}$ . 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) Beilinson-Bernstein localisation again: continued Notes

September 3 (2pm-4pm) Note early start time!

Yaping Yang (Melbourne) An introduction to the Riemann–Hilbert correspondence: continued Notes

Dougal Davis (Edinburgh) Beilinson-Bernstein localisation again Notes

Abstract: In this talk, I will go back over Beilinson-Bernstein localisation in a bit more detail. This time, I will cover the general definition of twisted differential operators for non-integral weights, and the closely related notion of monodromic D-modules. With the aid of some very explicit examples for $G = SL_2$ , I will also explain how to go back and forth between representations of the Lie algebra and monodromic D-modules on the flag variety in practice.

August 27 Gufang Zhao (Melbourne) Proof of Kazhdan-Lusztig 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 $X$ be an algebraic variety over $\mathbb{C}$ and let $X^{an}$ 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 $X$ with regular singularities and the category of local systems on $X^{an}$ . More generally, the Riemann–Hilbert correspondence for regular holonomic D-modules was proved by Kashiwara and Mebkhout independently in 1984. It says the de Rham functor induces an equivalence from the category of regular holonomic D-modules on $X$ to the category of perverse sheaves on $X^{an}$ .

August 20 Peter McNamara (Melbourne) Beilinson-Bernstein localisation: continued Notes

Gufang Zhao (Melbourne) Proof of Kazhdan-Lusztig conjectures on the Hecke algebra Notes

Abstract: In this talk, we recall Kazhdan-Lusztig conjectures on the Hecke algebra. Using the Beilinson-Bernstein localization theorem from Peter’s talk, and the Riemann-Hilbert 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 Kazhdan-Lusztig conjecture expressing the coefficients of irreducible modules with respect to Verma modules of a Lie algebra in terms of Hecke algebras, and show the non-negativity of the coefficients.

August 13 Peter McNamara (Melbourne) Beilinson-Bernstein 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 D-modules 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 Vershik-Okounkov approach.

Time: Wednesdays 13:15-14:15pm

Location: Zoom (Please contact the organisers for the link)

Schedule

3 Mar 13:15-14:15 Zhongtian Chen Universal positive self-adjoint Hopf algebra: uniqueness, continued.

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


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 10/17/24, 2021	Universal positive self-adjoint Hopf algebra: uniqueness.	[Ze] 3	Zhongtian Chen
	Universal positive self-adjoint Hopf algebra: special elements.	[Ze] 4	Ennes Mehmedbasic
	Symmetric functions, induction and restriction.	[Ze] 5.3, 6	Kshitija Vaidya
	Gelfand-Zeitlin algebra and Young-Jucys-Murphy 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, 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.