The invariant factors of the incidence matrices of points and subspaces in 𝑃𝐺(𝑛,𝑞) and 𝐴𝐺(𝑛,𝑞)

Chandler, David; Sin, Peter; Xiang, Qing

doi:10.1090/S0002-9947-06-03859-1

The invariant factors of the incidence matrices of points and subspaces in $\operatorname {PG}(n,q)$ and $\operatorname {AG}(n,q)$
HTML articles powered by AMS MathViewer

by David B. Chandler, Peter Sin and Qing Xiang PDF

Trans. Amer. Math. Soc. 358 (2006), 4935-4957 Request permission

Abstract:

We determine the Smith normal forms of the incidence matrices of points and projective $(r-1)$-dimensional subspaces of $\operatorname {PG}(n,q)$ and of the incidence matrices of points and $r$-dimensional affine subspaces of $\operatorname {AG}(n,q)$ for all $n$, $r$, and arbitrary prime power $q$.

References

M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
E. F. Assmus Jr. and J. D. Key, Designs and their codes, Cambridge Tracts in Mathematics, vol. 103, Cambridge University Press, Cambridge, 1992. MR 1192126, DOI 10.1017/CBO9781316529836
James Ax, Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964), 255–261. MR 160775, DOI 10.2307/2373163
Matthew Bardoe and Peter Sin, The permutation modules for $\textrm {GL}(n+1,\textbf {F}_q)$ acting on $\textbf {P}^n(\textbf {F}_q)$ and $\textbf {F}^{n-1}_q$, J. London Math. Soc. (2) 61 (2000), no. 1, 58–80. MR 1745400, DOI 10.1112/S002461079900839X
T. Beth, D. Jungnickel, H. Lenz, Design Theory, vol. 1, Second edition, Cambridge University Press, Cambridge, 1999.
S. C. Black and R. J. List, On certain abelian groups associated with finite projective geometries, Geom. Dedicata 33 (1990), no. 1, 13–19. MR 1042620, DOI 10.1007/BF00147596
D. B. Chandler, The Smith normal forms of designs with classical parameters, Ph.D. thesis, University of Delaware, 2004.
David B. Chandler and Qing Xiang, The invariant factors of some cyclic difference sets, J. Combin. Theory Ser. A 101 (2003), no. 1, 131–146. MR 1953284, DOI 10.1016/S0097-3165(02)00023-7
P. M. Cohn, Algebra, Vol. 1, John Wiley & Sons, London-New York-Sydney, 1974. MR 0360046
Bernard Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960), 631–648. MR 140494, DOI 10.2307/2372974
Avital Frumkin and Arieh Yakir, Rank of inclusion matrices and modular representation theory, Israel J. Math. 71 (1990), no. 3, 309–320. MR 1088823, DOI 10.1007/BF02773749
D. G. Glynn and J. W. P. Hirschfeld, On the classification of geometric codes by polynomial functions, Des. Codes Cryptogr. 6 (1995), no. 3, 189–204. MR 1351843, DOI 10.1007/BF01388474
C. D. Godsil, Problems in algebraic combinatorics, Electron. J. Combin. 2 (1995), Feature 1, approx. 20. MR 1312732, DOI 10.37236/1224
Noboru Hamada, The rank of the incidence matrix of points and $d$-flats in finite geometries, J. Sci. Hiroshima Univ. Ser. A-I Math. 32 (1968), 381–396. MR 243903
William M. Kantor, On incidence matrices of finite projective and affine spaces, Math. Z. 124 (1972), 315–318. MR 377681, DOI 10.1007/BF01113923
E. S. Lander, Topics in algebraic coding theory, D. Phil. Thesis, Oxford University, 1980.
R. Liebler, personal communication (2002).
Peter Sin, The elementary divisors of the incidence matrices of points and linear subspaces in $\mathbf P^n(\mathbf F_p)$, J. Algebra 232 (2000), no. 1, 76–85. MR 1783914, DOI 10.1006/jabr.2000.8387
K. J. C. Smith, Majority decodable codes derived from finite geometries, Mimeograph Series 561, Institute of Statistics, Chapel Hill, NC, 1967.
L. Stickelberger, Über eine Verallgemeinerung der Kreistheilung, Math. Annalen 37 (1890), 321–367.
Da Qing Wan, A Chevalley-Warning approach to $p$-adic estimates of character sums, Proc. Amer. Math. Soc. 123 (1995), no. 1, 45–54. MR 1215208, DOI 10.1090/S0002-9939-1995-1215208-3
Richard M. Wilson, A diagonal form for the incidence matrices of $t$-subsets vs. $k$-subsets, European J. Combin. 11 (1990), no. 6, 609–615. MR 1078717, DOI 10.1016/S0195-6698(13)80046-7