On Classification and Characterization
of Association Schemes:
C1. (with G. Bhattacharyya and R. Tanaka) Terwilliger Algebras of
Wreath Products of One-Class Association Schemes, Journal of Algebraic Combinatorics, DOI:10.1007/s10801-009-0196-x
C2. (with S. Bang) On Generalized Semidirect Product of
Association Schemes, Discrete
Mathematics, 303
(2005), 5 -- 16.
C3. (with S. Bang & M. Hirasaka) Semidirect
Products
of
Association Schemes, Journal of Algebraic Combinatorics, 22
No.1 (2005), 23-38.
There are many ways to construct new
association schemes from old ones. Association schemes can be built up from
`smaller' ones; two important constructive methods are the direct product and
wreath product. Another way to construct new association schemes from old is by
fusion and fission processes--the processes in which a new association scheme
is obtained by combining or splitting relations of the old scheme in a certain
way (cf. R3 below). In this paper, the semidirect product operation is
introduced as yet another way to construct new association schemes from smaller
ones. In his work developing the theory of association schemes as a
'generalized' group theory, Zieschang introduced the concept of the semidirect
product as a possible product operation of certain association schemes in 1996.
In 2000, Muzychuk generalized the Zieschang’s product operation slightly.
However, both Zieschang and Muzychuk’s operations are restricted to taking the
product of an association scheme with a ‘thin’ association scheme. In this paper we extend the semidirect
product operation into the entire set of association schemes. We then derive a
way to decompose certain association schemes into smaller association schemes.
We also investigate to what extent this product helps us to understand and
characterize the structure of association schemes. We give some examples to
show that the semidirect product produces many schemes that cannot be described
as neither the direct product nor the wreath product of smaller schemes.
C4. (with S. Bang) Characterization of Maximal Rational Circulant Association Schemes, Codes and
Designs, (eds. K. Arasu and A. Seress) dedicated to Dijen K.
Ray-Chaudhuri’s 65th Birthday, deGuyter,
An association scheme is circulant if its
relation matrix is equivalent to a circulant matrix. All fusion schemes of the
regular group scheme of a cyclic group are circulant. Every maximal rational
circulant association scheme arises as a fusion scheme of the regular group
scheme according to a certain standard partition of the cyclic group. In this
paper we give a complete description of the structure of maximal rational
circulant association schemes in terms of the direct and wreath products of
trivial association schemes of primie order.
C5. (with K. See) Association Schemes of
Small Order, Journal of Statistical Planning and Inference, 73
(1998) Nos. 1/2, 225-271
The association schemes of order up to 15
were classified mostly by the collaborative efforts of Nomiyama, Hirasaka, and
Suga in Bannai's school in 1994 - 1998. (Although there had been some sporadic
results obtained by others, their work has been developed independently.) In
this paper, we collect all isomorphism classes of association schemes of order
up to 15, and survey main tools that are useful for dealing with the
classification problem of association schemes of small order. Part of the paper
deals with the construction and enumeration of association schemes via Schur
rings and two ways of tensoring association matrices, and part deals with the
fusion and fission relations of association schemes of a given order by using
the notion of the wreath product and direct product of association schemes. The
association schemes are presented in Hasse diagrams of partially ordered sets
under the fusion relations.
Note: Hanaki and Miyamoto enumerated all the
isomorphism classes of association schemes up to order 19. (For order 16 &
17, see Kyushu J. Math. 52, (1998) No.2, 383-396, and for order 18 &
19, Korean J. Computational & Applied Math. 5, (1998) No.3,
543-552.) In October 1998, Miyamoto informed me that they enumerated symmetric
association schemes of order 23 and all primitive association schemes of order
up to 24. (The details are found at "http://math.shinshu-u.ac.jp/~hanaki".)
On Fusion and Fission of Association
Schemes:
F1. Class 3 Association Schemes Whose
Symmetrization Have Two Classes, Journal of Combinatorial Theory A
70 (1995) No.1, 1-29. [MR# 96b:05176; Zbl.Math.842.05099]
F2. Commutative Association Schemes Whose
Symmetrizations Have Two Classes, Journal of Algebraic Combinatorics
5 (1996), 47-55. [MR#96k:05213; Zbl.Math.843.05103].
F3. Fusion Relations in Products
of Association
Schemes, Graphs and Combinatorics. 18 (2002) 655-665.
F4. Fission Schemes of Pseudo Cyclic
Association Schemes (in preparation)
F5. (with L. K. Jorgensen, G. A. Jones, M. H. Klin) The Normally Regular Digraphs, Association Schemes and Related Combinatorial Structures (in preparation)
F6. (with F.
Adams, A. Jendreau, O.Olmez) Construction
of Directed Strongly Regular Graphs Using Regular Tournaments (in
preparation)
In these papers we investigated fusion and
fission relations in commutative association schemes in a systematic way by
studying their character tables, and tried to classify small class of
association schemes that have a specific fusion or fission pattern. Also, we
investigated fusion relations between the association schemes obtained as the
wreath product, direct product and semidirect product of various association
schemes. In this direction of research, it is natural to study the structure of
regular directed graphs appeared as the relation graphs of the schemes.
On Character Tables of Association
Schemes:
CT1. (with H. Tanaka) Group-Case Commutative Association
Schemes and Their Character Tables Proceedings of
Conference on Algebriac Combinatorics dedicating Eiichi Bannai's 60th birthday,
held in Sendai, Japan, June 2006.
CT2. (with
CT3. (with
CT4. (with
CT5. (with E. Bannai and
CT6. (with K. W. Johnson and J. D. H. Smith)
Characters of Finite Quasigroups VI: Critical Examples and Doubletons, European
Journal of Combinatorics, 11 (1990), 267-275. [MR#91f:20079; Zbl.Math.704.20056]
CT7. (with
CT8. (with
CT9. (with
CT10. (with
CT11. (with
These papers not only provide the character
tables of corresponding association schemes but also reveal many interesting
relationships between the character theory of association schemes and those of
groups, Moufang loops and quasigroups. In particular, they show that the
character tables of many classical groups and many permutation groups acting on
finite geometries can be obtained in a systematic way from the character tables
of corresponding association schemes.
On Block Designs:
D1 (With J. Hegeman, J. Langford, G. Bhattacharyya, and J. Kim) Some Existence and Construction Results of Polygonal Designs: European Journal of Combinatorics, 29 (2008), 1396-1407.
D2. (With K. See and J. Stufken) On a
Class of Partially Balanced Incomplete Block Designs with Applications in
Survey Sampling, Communications In Statistics Theory and Methods 1
(1997), 1-13. [MR#98f:62021]
D3. (with K. Driessel, K. See and J.
Stufken) Polygonal Designs: Some Existence and Non-existence Results, Journal
of Statistical Planning and Inference, 77 (1999), No.1, 155-166.
D4. (with A. Bailer, K.See and J. Stufken) Relative
Efficiencies of Sampling Plans for Selecting a Small Number of Units from a
Rectangular Region, Journal of
Statistical computation and Simulation, 66 No. 4 (2000), 273-294.
D5. (with K. See) Spatially Constrained
Sampling, In The Encyclopedia of Environmetrics (eds. A. El-Shaarawi and
We construct a class of balanced incomplete
block designs, called balanced sampling plans excluding contiguous units as a
continuation of the work initiated by Hedayat, Rao and Stufken in sampling
survey. We study on the structure of related designs through a combinatorial
enumeration method. We also investigate the efficiency and effectiveness of the
balanced sampling designs comparing with other conventional sampling designs
for fixed finite populations.
On Graphs:
G1. (with D. Eisenstat, J. Kim, D. Watson) Strongly Regular
Graphs with Parameters (64, 28, 12, 12) (in preparation)
G2. Products of Distance Regular Graphs,
Utilitas Mathematica 29 (1986), 173-175 [MR#87g:05209]
G3. (F5) (with L. K. Jorgensen, G. A. Jones, M. H. Klin) The Normally Regular Digraphs,
Association Schemes and Related Combinatorial Structures (in preparation)
Miscellany:
M1. (with C. Godsil) Association Schemes, in: C. Colbourn and
J. Dinitz (Eds.). The CRC Handbook of
COMBINATORIAL DESIGNS (2nd. Ed.), CRC Press inc., Boca Raton, 2007,
325-330.
M2. (with D. Choi) Spanning Time for
Abstract Biological Codes, Proceedings of Hungarian-Korean Combinatorics
Workshop held in June 2001, at Renyi Institute of Mathematics in
M3. Posets Related to Some Association
Schemes (unpublished preprint)
M4. Commutative Association Schemes and
Related Algebra, Proceedings of the Fifth KIT Mathematics Workshop,
"Korea Institute of Technology, Korea, August 1990, 5, 143-173.
[MR#92d:05184]
M5. Commutative Association Schemes and
Their Fusion and Fission Collected Papers Dedicated to Professor Yeonsik
Kim on the Occasion the 60th Birthday, Ed. I. Jeong, et al., University
publishers, Korea, 1992, 113-130.
M6. The Terwilliger Algebras of Wreath
Product Association Schemes (in preparation)
M7. (with K. See) An approximation with
Multilinear Models for
M8. (with M. Maxwell) Characterization of
distance-regular graphs obtained from almost perfect nonlinear functions (in
preparation)
M9. (with J. Kim) Classification of small
class association schemes of order 64 (in preparation)
M10. Arithmetic in the Jacobian of a hyperelliptic
curve.(in preparation)
M11. Almost perfect nonlinear functions and
almost bent functions (in preparation)
Lecture Notes:
* Lecture notes for the courses that I have
taught at ISU
L1. (M605) Designs, Codes, and
Association Schemes
L2. (M606) Topics in Combinatorics: Finite
Sets, Extremal Set Systems, and Relations
L3. (452X) Discrete Mathematics for
Information Assurance
L4. (M504-505) Abstract Algebra: Groups,
Rings, and Fields.
L5. (M533) Introduction to
Cryptography