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Hermitian Varieties in a Finite Projective Space PG(N, q2)

Published online by Cambridge University Press:  20 November 2018

R. C. Bose  and
I. M. Chakravarti
R. C. Bose
Affiliation:
University of North Carolina
I. M. Chakravarti
Affiliation:
University of North Carolina
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The geometry of quadric varieties (hypersurfaces) in finite projective spaces of N dimensions has been studied by Primrose (12) and Ray-Chaudhuri (13). In this paper we study the geometry of another class of varieties, which we call Hermitian varieties and which have many properties analogous to quadrics. Hermitian varieties are defined only for finite projective spaces for which the ground (Galois field) GF(q2) has order q2, where q is the power of a prime. If h is any element of GF(q2), then = hq is defined to be conjugate to h.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 1161 - 1182
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Baer, R., Linear algebra and projective geometry (New York, 1952).Google Scholar
2. Bose, R. C., On the construction of balanced incomplete block designs, Ann. Eugen. London, 9 (1939), 358–399.Google Scholar
3. Bose, R. C., On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements, Golden jubilee commem. vol., Cal. Math. Soc. (1958-59), 341–354.Google Scholar
4. Bose, R. C., Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math., 13 (1963), 389–419.Google Scholar
5. Bose, R. C., Combinatorial properties of partially balanced designs and association schemes, Sankhya, ser. A, 25 (1963), 109–136; and Contributions to statistics, presented to Professor P. C. Mahalanobis on the occasion of his 70th birthday (Calcutta, 1964), 21-48.Google Scholar
6. Bose, R. C. and Clatworthy, W. H., Some classes of partially balanced designs, Ann. Math. Statist., 26 (1955), 212–232.Google Scholar
7. Bose, R. C. and Mesner, D. M., On linear associative algebras corresponding to association schemes of partially balanced designs, Ann. Math. Statist., 30 (1959), 21–38.Google Scholar
8. Bose, R. C. and Nair, K. R., Partially balanced incomplete block designs, Sankhya, 4 (1939), 337–372.Google Scholar
9. Bose, R. C. and Shimamoto, T., Classification and analysis of partially balanced designs with two associate classes, J. Amer. Statist. Assoc, 47 (1952), 151–184.Google Scholar
10. Carmichael, R., Introduction to the theory of groups of finite order (New York, 1956), chaps. XI and XII.Google Scholar
11. Mann, H. B., Analysis and design of experiments (New York, 1949). chap. IX.Google Scholar
12. Primrose, E. J. F., Quadrics infinite geometries, Proc. Cambridge Philos. Soc., 47 (1951), 299–304.Google Scholar
13. Ray-Chaudhuri, D. K., Some results on quadrics infinite projective geometries based on Galois fields, Can. J. Math., 14 (1962), 129–138.Google Scholar
14. Ray-Chaudhuri, D. K., Application of the geometry of quadrics for constructing PBIB designs, Ann. Math. Statist., 33 (1962), 1175–1186.Google Scholar
15. Segre, B., Lectures on modern geometry (Rome, 1960). chap. 17.Google Scholar
16. Shrikhande, S. S. and Singh, N. K., On a method of constructing symmetrical balanced incomplete block designs, Sankhya, ser. A, 24 (1962), 25–32.Google Scholar
17. Takeuchi, K., On the construction of a series of BIB designs, Statist. Appl. Res., JUSE, 10 (1963), 48.Google Scholar