Abstract
A brief introduction is given to the topic of Smith normal forms of incidence matrices. A general discussion of techniques is illustrated by some classical examples. Some recent advances are described and the limits of our current understanding are indicated.
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Arslan O, Sin P. Some simple modules for classical groups and p-ranks of orthogonal and Hermitian geometries. J Algebra, 2011, 327: 141–169
Bardoe M, Sin P. The permutation modules for GL(n+1,Fq) acting on Pn(Fq) and F n+1q . J London Math Soc, 2000, 61: 58–80
Berndt B C, Evans R J, Williams K S. Gauss and Jacobi Sums. In: Canadian Mathematical Society Series of Monographs and Advanced Texts. New York: John Wiley & Sons Inc., 1998
Bier T. Remarks on recent formulas of Wilson and Frankl. European J Combin, 1993, 14: 1–8
Bittner S, Guo X, Zweber A. Approaches to Rota’s basis conjecture. Report on James Madison University Summer REU, http://educ.jmu.edu/duceyje/reu/report2.pdf, 2012
Black S C, List R J. On certain abelian groups associated with finite projective geometries. Geom Dedicata, 1990, 33: 13–19
Brouwer A E. The eigenvalues of oppositeness graphs in buildings of spherical type. In: Combinatorics and Graphs, vol. 531. Providence, RI: Amer Math Soc, 2010, 1–10
Brouwer A E, Ducey J E, Sin P. The elementary divisors of the incidence matrix of skew lines in PG(3, q). Proc Amer Math Soc, 2012, 140: 2561–2573
Brouwer A E, Haemers W H. Association Schemes. In: Handbook of Combinatorics, vol. 1, 2. Amsterdam: Elsevier, 1995, 747–771
Brouwer A E, Haemers W H. Spectra of Graphs. New York: Springer, 2012
Chandler D B, Sin P, Xiang Q. The invariant factors of the incidence matrices of points and subspaces in PG(n, q) and AG(n, q). Trans Amer Math Soc, 2006, 358: 4935–4957
Chandler D B, Xiang Q. The invariant factors of some cyclic difference sets. J Combin Theory Ser A, 2003, 101: 131–146
Chandler D B. The Smith Normal Forms of Designs with Classical Parameters. Ann Arbor, MI: ProQuest LLC, 2004
Delsarte P. An Algebraic Approach to the Association Schemes of Coding Theory. In: Philips Research Reports Supplements, vol. 10. Amsterdam: N V Philips’ Gloeilampenfabrieken, 1973
Evans R, Hollmann H, Krattenthaler C, et al. Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets. J Combin Theory Ser A, 1999, 87: 74–119
Frankl P. Intersection theorems and mod p rank of inclusion matrices. J Combin Theory Ser A, 1990, 54: 85–94
Hamada N. The rank of the incidence matrix of points and d-flats in finite geometries. J Sci Hiroshima Univ Ser A-I Math, 1968, 32: 381–396
Hiss G. Hermitian function fields, classical unitals, and representations of 3-dimensional unitary groups. Indag Math N S, 2004, 15: 223–243
James G D. Representations of general linear groups. In: London Mathematical Society Lecture Note Series, vol. 94. Cambridge: Cambridge University Press, 1984
James G, Kerber A. The representation theory of the symmetric group. In: Encyclopedia of Mathematics and its Applications, vol. 16. Reading, MA: Addison-Wesley Publishing Co., 1981
Jantzen J C. Representations of algebraic groups, Pure and Applied Mathematics, vol. 131. Boston, MA: Academic Press Inc., 1987
Lander E S. Topics in Algebraic Coding Theory. Oxford: University of Oxford, 1980
Lander E S. Symmetric designs: an algebraic approach. In: London Mathematical Society Lecture Note Series, vol. 74. Cambridge: Cambridge University Press, 1983
Lataille J M. The elementary divisors of incidence matrices between certain subspaces of a finite symplectic space. J Algebra, 2003, 268: 444–462
MacWilliams F J, Mann H B. On the p-rank of the design matrix of a difference set. Inform Control, 1968, 12: 474–488
Payne S E, Thas J A. Finite Generalized Quadrangles, 2nd ed. In: EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society Publishing House, 2009
Queiró J F. Axioms for invariant factors. Portugal Math, 1997, 54: 263–269
Rushanan J J. Combinatorial applications of the Smith normal form. Congr Numer, 1990, 73: 249–254
Rushanan J J. Topics in Integral Matrices and Abelian Group Codes. Ann Arbor, MI: ProQuest LLC, 1986
Sin P. Oppositeness in buildings and simple modules for finite groups of lie type. Buildings, Finite Geometries and Groups, 2012, 10: 273–286
Sin P. The p-rank of the incidence matrix of intersecting linear subspaces. Des Codes Cryptogr, 2004, 31: 213–220
Sin P, Tiep P H. Rank 3 permutation modules of the finite classical groups. J Algebra, 2005, 291: 551–606
Sin P, Wu J H, Xiang Q. Dimensions of some binary codes arising from a conic in PG(2, q). J Combin Theory Ser A, 2011, 118: 853–878
Smith H J S. Arithmetical notes. Proc London Math Soc, 1873, 4: 236–253
Wan D Q. A Chevalley-Warning approach to p-adic estimates of character sums. Proc Amer Math Soc, 1995, 123: 45–54
Wilson R M. A diagonal form for the incidence matrices of t-subsets vs. k-subsets. European J Combin, 1990, 11: 609–615
Wilson R M, Wong T W H. Diagonal forms of incidence matrices associated with t-uniform hypergraphs. Preprint, 2012
Wong T W H. Diagonal forms and zero sum (mod 2) bipartite ramsey numbers. Preprint, 2012
Wu J H. Geometric Structures and Linear Codes Related to Conics in Classical Projective Planes of Odd Orders. Ann Arbor, MI: ProQuest LLC, 2008
Wu J H. Some p-ranks related to a conic in PG(2, q). J Combin Design, 2010, 18: 224–236
Xiang Q. Recent results on p-ranks and Smith normal forms of some 2-(v, k, λ) designs. In: Coding Theory and Quantum Computing, Contemp Math, vol. 381. Providence, RI: Amer Math Soc, 2005, 53–67
Yamamoto K. On congruences arising from relative Gauss sums. In: Number Theory and Combinatorics. Singapore: World Scientific Publishing, 1985, 423–446
Yamamoto S, Fujii Y, Hamada N. Composition of some series of association algebras. J Sci Hiroshima Univ Ser A-I Math, 1965, 29: 181–215
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Sin, P. Smith normal forms of incidence matrices. Sci. China Math. 56, 1359–1371 (2013). https://doi.org/10.1007/s11425-013-4643-8
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DOI: https://doi.org/10.1007/s11425-013-4643-8