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), 255β261. 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.Β 1, 58β80. 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.Β 1, 13β19. 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.Β 1, 131β146. 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), 631β648. 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.Β 3, 309β320. 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.Β 3, 189β204. 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), 381β396. MR 243903
- William M. Kantor, On incidence matrices of finite projective and affine spaces, Math. Z. 124 (1972), 315β318. 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.Β 1, 76β85. 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, Γber eine Verallgemeinerung der Kreistheilung, Math. Annalen 37 (1890), 321β367.
- Da Qing Wan, A Chevalley-Warning approach to $p$-adic estimates of character sums, Proc. Amer. Math. Soc. 123 (1995), no.Β 1, 45β54. 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.Β 6, 609β615. MR 1078717, DOI 10.1016/S0195-6698(13)80046-7
Additional Information
- David B. Chandler
- Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
- Address at time of publication: Institute of Mathematics, Academia Sinica, NanGang, Taipei 11529, Taiwan
- Email: chandler@math.udel.edu
- Peter Sin
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
- Email: sin@math.ufl.edu
- Qing Xiang
- Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
- Email: xiang@math.udel.edu
- Received by editor(s): April 27, 2004
- Received by editor(s) in revised form: September 27, 2004
- Published electronically: April 11, 2006
- Additional Notes: The second author was partially supported by NSF grant DMS-0071060. The third author was partially supported by NSA grant MDA904-01-1-0036.
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 4935-4957
- MSC (2000): Primary 05E20; Secondary 20G05, 20C11
- DOI: https://doi.org/10.1090/S0002-9947-06-03859-1
- MathSciNet review: 2231879