Algebraic Combinatorics School 2015

The Algebraic Combinatorics School 2015 (ACS2015) will be held at Korea Institute for Advanced Study (KIAS), Seoul on February 10-13, 2015.

The purpose of this school is not just presenting individual works but teaching basic concepts and tools of algebraic combinatorics. So we invite 3 lecturers. The style of all talks tends to be less formal.

In principle, registration is on a first-come, first-served basis. If you want to participate, please write a registration form until January 31.

Information

  • Title Algebraic Combinatorics School 2015 (대수적 조합수학 스쿨 2015)
  • Date February 10-13 (Tue-Fri), 2015
  • Venue KIAS, Seoul
    • Room 1114 in Feb. 10
    • Room 1503 in Feb. 11~13
  • Web http://event.combinatorics.kr/acs2015
  • Organizers
    • Jang Soo Kim (Sungkyunkwan University)
    • Kyungyong Lee (Wayne State University and KIAS)
    • Seunghyun Seo (Kangwon National University)
    • Heesung Shin (Inha University)
  • Host / Sponsor KIAS
  • We are going to
    • give twelve 60-minute lectures in Korean.
    • provide food and (limited) accommodations for all participants who register until January 31(Participants not from Seoul or nearby cities have priority.)
    • distribute the abstracts of ACS2015.

Invited Lecturers

  • Soojin Cho (Ajou University)
  • Jae-Hoon Kwon (Sungkyunkwan University)
  • Kyungyong Lee (Wayne State University and KIAS)

    Timetable

    • February 10 (Tuesday)
      • 00h00 - 14h00 Registration
      • 14h00 - 00h00 Opening Ceremony
      • 14h00 - 15h00 Lecture K-1
      • 15h30 - 16h30 Lecture C-1
      • 17h00 - 18h00 Lecture L-1
      • 18h00 - 00h00 Dinner
    • February 11 (Wednesday)
      • 08h30 - 09h30 Breakfast
      • 09h30 - 10h30 Lecture L-2
      • 11h00 - 12h00 Lecture C-2
      • 12h00 - 14h00 Lunch
      • 14h00 - 15h00 Lecture K-2
      • 15h30 - 16h30 Discussion 1
      • 17h00 - 18h00 Discussion 2
      • 18h00 - 00h00 Dinner (Banquet)
    • February 12 (Thursday)
      • 08h30 - 09h30 Breakfast
      • 09h30 - 10h30 Lecture K-3
      • 11h00 - 12h00 Lecture C-3
      • 12h00 - 14h00 Lunch
      • 14h00 - 15h00 Lecture L-3
      • 15h30 - 16h30 Discussion 3
      • 17h00 - 18h00 Discussion 4
      • 18h00 - 00h00 Dinner
    • February 13 (Friday)
      • 08h30 - 09h30 Breakfast
      • 09h30 - 10h30 Lecture K-4
      • 11h00 - 12h00 Lecture C-4
      • 12h00 - 14h00 Lunch
      • 14h00 - 15h00 Lecture L-4
      • 15h00 - 15h10 Closing Ceremony
      • 15h30 - 00h00 Discussion 5

    Program

    • Lecture C
      • Speaker Soojin Cho (Ajou University)
      • Title Combinatorics of Coxeter Groups; a basic
      • Abstract Coxeter group을 정의하고, 다음 관련주제에 관한 기본이론을 A type을 중심으로 소개한다.
        • Bruhat order
        • Reduced decompositions
        • Stanley Symmetric functions
        • Quasi-symmetric functions
      • References
        1. Björner, Anders; Brenti, Francesco “Combinatorics of Coxeter groups.” Graduate Texts in Mathematics, 231. Springer, New York, 2005
        2. Sagan, Bruce E. “The symmetric group. Representations, combinatorial algorithms, and symmetric functions.” Second edition. Graduate Texts in Mathematics, 203. Springer-Verlag, New York, 2001
        3. Stanley, Richard P. “Enumerative combinatorics.” Vol. 2. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999.
      • Lectures
        • C-1 Symmetric functions
          We first introduce the ‘Coxeter systems’ and nice properties that they have; exchange property and deletion property. Then, we consider a poset structure on Coxeter groups, called (weak) Bruhat order with some combinatorial examples at least for type An.
        • C-2 & C-3 Quasi-symmetric functions and Stanley Symmetric functions
          The notion of ‘quasi-symmetric functions’ will be introduced. Then we will see how Schur functions are written as sums of ‘fundamental’ quasi-symmetric functions, in which the statistics ‘descent’ on the symmetric group (Coxeter group of type A) plays a role.
          Stanley symmetric functions were defined to enumerate the reduced expressions of elements in Coxeter groups: We give the definition of Stanley symmetric functions in terms of quasi symmetric functions, and see how the theory of symmetric functions is used to count the number of reduced expressions. Edelman-Greene correspondence which combinatorially shows that Stanley symmetric functions are positive sums of Schur functions will also be mentioned. We consider Coxeter groups of other types in this context also, if time permits.
        • C-4 Poincaré series and Eulerian polynomials
          We look at generating functions of statistics ‘length’, ‘descent’, and both on Coxeter groups. Recursion formulae and some nice properties they satisfy will be discussed.

    • Lecture K
      • Speaker Jae-Hoon Kwon (Sungkyunkwan University)
      • Title Introduction to symmetric functions and representation theory
      • Abstract In this lecture, we give an introduction to the theory of symmetric functions in connection with representations of symmetric groups and general linear groups. It is intended for those who are interested in these topics but not yet familiar with them. The lecture is divided into the following:
        • Symmetric functions
        • Combinatorics of Young tableaux
        • Representations of symmetric groups
        • Representations of general linear groups
        These topics will (hopefully) cover very classical results on combinatorial representation theory and algebraic combinatorics, which are necessary background to understand modern research topics like representations of quantum groups and Hecke algebras, categorification and so on.
      • References
        1. I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 2nd ed., 1995. (Chapter 1)
        2. W. Fulton, Young tableaux, London Mathematical Society Student Texts, 35. Cambridge University Press, 1997.(Parts I and II)
        3. R. Stanley, Enumerative Combinatorics, Vol 2, Cambridge University Press, 1999. (Chapter 7 and Appendix A)
        4. Slides
      • Lectures
        • K-1 Symmetric functions / Slide
          We introduce the notion of symmetric functions and review some of its basic properties. We study linear bases of the ring of symmetric functions including a basis of Schur functions, the Hall inner product, and its Hopf algebra structure.
        • K-2 Combinatorics of Young tableaux / Slide
          We introduce a combinatorial algorithm for Young tableaux, which is called the Schensted's bumping. Based on this, we study the celebrated Robinson-Schensted-Knuth (simply RSK) correspondence and its various applications. We also study the Littlewood-Richardson coefficients, which give multiplicative structure constants for the ring of symmetric functions with respect to Schur functions.
        • K-3 Representations of symmetric groups / Slide
          We briefly review some basic background on representation theory of finite groups. We then explain how the representation theory of symmetric groups over complex numbers is related with the theory of symmetric functions. More precisely, we will see that the character ring of symmetric groups is isomorphic to the ring of symmetric functions and in particular the irreducible characters of symmetric groups are completely determined by a transition matrix between two linear bases of the ring of symmetric functions.
        • K-4 Representations of general linear groups / Slide
          We review well-known results on complex finite-dimensional representations of general linear groups, and then explain its connection with the theory of symmetric functions. We give representation theoretic interpretations or applications to representation theory of combinatorial results in Lectures 1 and 2. If time permits, we explain two important duality theorems: Schur-Weyl duality, which connects the representations of symmetric groups and general linear groups, and Howe duality, which underlies the RSK correspondence.

    • Lecture L
      • Speaker Kyungyong Lee (Wayne State University and KIAS)
      • Title Cluster algebras and related combinatorics
      • Abstract Cluster algebras were discovered by Fomin and Zelevinsky in 2001. Since then, they are shown to be related to many branches of mathematics and physics. With very active development in the last decade, they become fundamental objects.
        A cluster algebra is a commutative algebra with distinguished generators called cluster variables. These cluster variables are Laurent polynomials obtained from highly nontrivial recursive relations. It is a very important problem to find combinatorial formulas for these Laurent expressions. These formulas will lead to lots of applications in algebra, geometry, topology, analysis and physics as well as combinatorics.
        We explain some known combinatorial formulas for certain special cases including the rank 2 case and the ones coming from Riemann surfaces. Along the way, we explore a number of interesting combinatorial objects : snake graphs, perfect matchings, Dyck paths and compatible pairs. We focus on very explicit computations for such objects.
      • Lectures
        • L-1 Introduction to cluster algebras
          A cluster algebra is a commutative algebra with distinguished generators called cluster variables. These cluster variables are constructed from certain recursive relations. To each directed graph (quiver), we define the associated cluster algebra. We also prove that the cluster variables are Laurent polynomials.
        • L-2 Cluster algebras coming from discs
          We explain how cluster algebras arise from triangulations of discs with marked points on the boundary. In this case we give combinatorial formulas for the Laurent expressions of cluster variables in terms of T-paths and globally compatible collections on Dyck paths.
        • L-3 Cluster algebras coming from Riemann surfaces
          We explain how cluster algebras arise from triangulations of Riemann surfaces with marked points and (possibly empty) boundaries. In this case we give combinatorial formulas for the Laurent expressions of cluster variables in terms of snake graphs and perfect matchings.
        • L-4 Cluster algebras of rank 2
          We define noncommutative cluster algebras of rank 2, using Kontsevich automorphisms. In this case we give a combinatorial formula for cluster variables in terms of Dyck paths and compatible pairs.

        Registration

        Discussion

        • List of participants who have solved the problems in discussion sessions.
          • Yeon-jae Hong
          • Byunghak Hwang
          • Byungyun Jeon
          • Bochan Jeong
          • Byeori Kim
          • Younghun Kim
          • Hyoje Lee
          • Junwoo Oh
          • Jihye Park
          • Kyoungsuk Park
          • Hyung-Ki Yoo
          • Sanghoon Yu

        Participants

        Showing 85 items
        English NameOriginal NameAffiliationRef.
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        English NameOriginal NameAffiliationRef.
        Kyungyong Lee 이경용 Wayne State University / KIAS Organizer & Lecturer 
        Jang Soo Kim 김장수 Sungkyunkwan University Organizer 
        Seunghyun Seo 서승현 Kangwon National University Organizer 
        Heesung Shin 신희성 Inha University Organizer 
        Jae-Hoon Kwon 권재훈 Sungkyunkwan University Lecturer 
        Soojin Cho 조수진 Ajou University Lecturer 
        Byeori Kim ε 김벼리 POSTECH Discussant 
        Younghun Kim ε 김영훈 Sogang University Discussant 
        Kyoungsuk Park ε 박경숙 Ajou University Discussant 
        Jihye Park ε 박지혜 Yeungnam University Discussant 
        Junwoo Oh ε 오준우 Seoul National University Discussant 
        Sanghoon Yu ε 유상훈 Seoul National University Discussant 
        Hyung-Ki Yoo ε 유형기 Chungnam National University Discussant 
        Hyoje Lee ε 이효제 Ajou University Discussant 
        Byungyun Jeon ε 전병연 Ajou University Discussant 
        Bochan Jeong ε 정보찬 Ajou University Discussant 
        Yeon-jae Hong ε 홍연재 Sungkyunkwan University Discussant 
        Byunghak Hwang ε 황병학 Seoul National University Discussant 
         김명호 KIAS A registrant on site 
         김종락 Sogang University A registrant on site 
         김현규 KIAS A registrant on site 
        Yoomi Rho 노유미 University of Incheon A registrant on site 
         박의용 University of Seoul A registrant on site 
         손재범 Yonsei University A registrant on site 
         신재호 NIMS A registrant on site 
        Sang-il Oum 엄상일 KAIST A registrant on site 
        Sang June Lee 이상준 Duksung Women's University A registrant on site 
         정지혜 Seoul National University A registrant on site 
        Kyounglo Kim ε 김경로 KAIST  
        Minki Kim ε 김민기 KAIST  
        Byungchan Kim 김병찬 SeoulTech  
        Sangjib Kim 김상집 Ewha Womans University  
        Sunah Kim ε 김선아 Seoul National University  
        Seonhwa Kim 김선화 IBS Center for Geometry and Physics  
        Sooyeong Kim ε 김수영 Sungkyunkwan University  
        Younjin Kim 김연진 KAIST  
        Yoongi Kim ε 김윤기 KAIST  
        Eunmi Kim 김은미 NIMS  
        Ilhyung Kim 김일형 Seoul National University  
        Jiwon Kim ε 김지원 Seoul National University  
        Jinha Kim ε 김진하 Seoul National University  
        Hana Kim 김하나 NIMS  
        Hoil Kim 김호일 Kyungbook National University  
        Sunyoung Nam ε 남선영 Sogang University  
        Sook Min 민숙 Yonsei University  
        Boram Park 박보람 Ajou University  
        Youngja Park 박영자 Yonsei University  
        Junyong Park ε 박준용 Sogang University  
        JinGoo Park ε 박진구 Sogang University  
        Younghan Bae ε 배영한 Seoul National University  
        Jineon Baek ε 백진언 POSTECH  
        Geewon Suh ε 서기원 KAIST  
        UhiRinn Suh 서의린 Seoul National University  
        U-Keun Song ε 송우근 Sungkyunkwan University  
        Hosang Song ε 송호상 Yonsei University  
        Hyeongdae Yang ε 양형대 Kyungpook National University  
        Se-jin Oh 오세진 Seoul National University  
        Semin Yoo ε 유세민 Ewha Womans University  
        Sun-mi Yun ε 윤선미 Sungkyunkwan University  
        Nari Lee ε 이나리 Sogang University  
        Dong-gyue Lee ε 이동규 Ajou University  
        Sanha Lee ε 이산하 Sungkyunkwan University  
        Sojeong Lee ε 이소정 Seoul National University  
        Seung Jin Lee 이승진 KIAS  
        Seungchul Lee ε 이승철 Seoul National University  
        Seunghun Lee ε 이승훈 KAIST  
        Jeong Hyeon Lee ε 이정현 KAIST  
        Sung Woo Chang ε 장성우 KAIST  
        Ahream Jang ε 장아름 Chungnam National University  
        Jinse Jang ε 장진세 Chungnam National University  
        Hyunmee Jun ε 전현미 Kyonggi University  
        Keehoon Jung ε 정기훈 Yonsei University  
        Ji-Hwan Jung 정지환 Sungkyunkwan University  
        Changsung Jo ε 조창성 Seoul National University  
        Sung-Tae Jin ε 진성태 Sungkyunkwan University  
        Seung-Il Choi ε 최승일 Sogang University  
        Ilkyoo Choi 최일규 KAIST  
        Jihyun Choi ε 최지현 Ewha Womans University  
        Jihoon Choi ε 최지훈 Seoul National University  
        Hanul Choi ε 최하늘 Chungnam National University  
        Hojin Choi ε 최호진 KAIST  
        Daehyun Ha ε 하대현 Pusan National University  
        DongMan Heo ε 허동만 Sogang University  
        Jaeseong Heo 허재성 Hanyang University  
        DongSeon Hwang 황동선 Ajou University  
        Showing 85 items