Fourth Korea - Japan Workshop on Algebra and Combinatorics ( sponsored by BK 21 - CoDiMaRO , PMI , KOSEF ,

Jin Ho Kwak (POSTECH) Title: Enumeration and genus distribution of maps on surfaces Abstract: Two 2-cell embeddings ı : X → S and  : X → S of a connected graph X into a closed orientable surface S are congruent if there are an orientation-preserving surface homeomorphism h on S and a graph automorphism γ of X such that ıh = γ. When we restrict γ as the identity, we say two embeddings are equivalent. A 2-cell embedding ı : X → S of a graph X into a closed orientable surface S is sometimes described combinatorially by a pair (X; ρ) called a map, where ρ is a product of disjoint cycle permutations each of which is the permutation of the dart set of X initiated at the same vertex following the orientation of S. First we review the enumerating the equivalence or congruence classes of embeddings of a graph and also their genus distribution problem. The mirror image of a map (X; ρ) is the map (X; ρ−1), and one of the corresponding embeddings is called the mirror image of the other. A 2-cell embedding of X is reflexible if it is congruent to its mirror image. Mull et al. [Proc. Amer. Math. Soc. 103(1988) 321330] developed an approach for enumerating the congruence classes of 2-cell embeddings of graphs into closed orientable surfaces. In this talk we also introduce a method for enumerating the congruence classes of reflexible 2-cell embeddings of graphs into closed orientable surfaces, and apply it to the complete graphs, the bouquets of circles, the dipoles and the wheel graphs to count their congruence classes of reflexible or nonreflexible (called chiral) embeddings. Paul Terwilliger (Wisconsin and Kanazawa University) Title: A Drinfel’d polynomial for tridiagonal pairs Abstract: We introduce a polynomial invariant for tridiagonal pairs called the Drinfel’d polynomial. We discuss how this polynomial generalizes the classical Drinfel’d polynomial associated with the loop algebra for sl2 or Uqsl2). We compute the roots of the Drinfel’d polynomial for the case in which the associated tridiagonal pair is a Leonard pair. This is joint work with Tatsuro Ito. YoungJu Choie (POSTECH) Title: Broue-Enguehard maps and Atkin-Lehner involutions Abstract: Let a be one of the ten integers such that the sum of their divisors divide 24. For each sucha, (except 15) we give a map from an algebra of polynomial invariants of some finite group to the algebra of modular forms invariant under the Atkin-Lehner group of level a. These maps are motivated and inspired by constructions of modular lattices from self-dual codes over rings. This work generalizes Broue-Enguehard work in level one and three obtained from binary and ternary codes. This is joint work with P.Solé Masaaki Harada (Yamagata University) Title: Two recent results in extremal self-dual codes Abstract: In my talk, I present two recent results on the existence and the classification of extremal self-dual codes. The existence of binary extremal doubly even self-dual codes was known for the following lengths 8, 16, 24, 32, 40, 48, 56, 64, 80, 88, 104 and 136. The first example of a binary