## Algebraic Combinatorics School 2015
The 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
. (Participants not from Seoul or nearby cities have priority.)**January 31** - 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**- Björner, Anders; Brenti, Francesco “Combinatorics of Coxeter groups.” Graduate Texts in Mathematics, 231. Springer, New York, 2005
- Sagan, Bruce E. “The symmetric group. Representations, combinatorial algorithms, and symmetric functions.” Second edition. Graduate Texts in Mathematics, 203. Springer-Verlag, New York, 2001
- Stanley, Richard P. “Enumerative combinatorics.” Vol. 2. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999.
**Lectures**
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 A__C-1 Symmetric functions__
_{n}.
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.__C-2 & C-3 Quasi-symmetric functions and Stanley Symmetric functions__
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.
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.*C-4 Poincaré series and Eulerian polynomials*
- 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
**References**- I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 2nd ed., 1995. (Chapter 1)
- W. Fulton, Young tableaux, London Mathematical Society Student Texts, 35. Cambridge University Press, 1997.(Parts I and II)
- R. Stanley, Enumerative Combinatorics, Vol 2, Cambridge University Press, 1999. (Chapter 7 and Appendix A)
- Slides
**Lectures**
/ Slide__K-1 Symmetric functions__ 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**
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-1 Introduction to cluster algebras__
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-2 Cluster algebras coming from discs__
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-3 Cluster algebras coming from Riemann surfaces__
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.__L-4 Cluster algebras of rank 2__
## Registration- There is NO registration fee. Just write a
until January 31.**registration form**
## 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 |

### ACS2015

English Name | Original Name | Affiliation | Ref. |
---|---|---|---|

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 |

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