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.

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

• • • 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 A_n.

• • • • 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.

• • • 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

• • • 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.

• 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

• • • 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

• There is NO registration fee. Just write a registration form until January 31.

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