An Analogue of Nakayama's Conjecture for
Johnson Schemes

Osamu Shimabukuro

Graduate School of Mathematics, Kyushu University 33, Fukuoka 812-8581, Japan

shima@math.kyushu-u.ac.jp

Annals of Combinatorics 9 (1) p.101-115 March, 2005

Abstract:

Nakayama's Conjecture is one of the most famous theorems for representation theory
of symmetric groups. Two general irreducible characters of a symmetric group belong to the
same *p*-block if and only if the *p*-cores of the young diagrams corresponding to them are the
same. The conjecture was first proven in 1947 by Brauer and Robinson. We consider an analogue
of Nakayama's Conjecture for Johnson scheme.

References:

1. E. Bannai and T. Ito, Algebraic Combinatorics I, Association Schemes, Benjamin, 1984.

2. P.J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1994.

3. P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. 10 (1973).

4. P. Delsarte, Properties and applications of the recurrence *F(i+1;k+1;n+1) = q ^{k+1}F(i;k+
1;n)–q^{k}F(i, k, n)*, SIAM J.Appl. Math. 31 (2) (1976) 262--270.

5. A. Hanaki, Block decomposition of standard modules, 2002, preprint, http://math.shinshu-u.ac.jp/%7Ehanaki/notes.html.

6. A. Hanaki, The number of irreducible modular representation of Johnson scheme (Japanese), preprint.

7. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl. 16, Addison - Wesley, 1981.

8. H. Nagao and Y. Tsushima, Representations of Finite Groups (Japanese), Shoukabou, 1987.