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Báo cáo toán học: "Finite vector spaces and certain lattices"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Finite vector spaces and certain lattices. | Finite vector spaces and certain lattices Thomas W. Cusick 106 Diefendorf Hall Department of Mathematics State University of New York at Buffalo Buffalo NY 14214-3093 E-mail cusick@acsu.buffalo.edu Submitted January 6 1998 Accepted March 18 1998 Abstract The Galois number Gn q is defined to be the number of subspaces of the n-dimensional vector space over the finite field GF q . When q is prime we prove that Gn q is equal to the number Ln q of n-dimensional mod q lattices which are defined to be lattices that is discrete additive subgroups of n-space contained in the integer lattice Zn and having the property that given any point P in the lattice all points of Zn which are congruent to P mod q are also in the lattice. For each n we prove that Ln q is a multiplicative function of q. Keywords Multiplicative function Lattice Galois numbers Vector space Identities 1991 Mathematical Reviews subject numbers Primary 05A15 05A19 11A25 11H06 Secondary 05A30 94A60 11T99 THE ELECTRONIC .JOURNAL OF COmBINATORICS 5 1998 R17 2 1 Introduction The well known Gaussian coefficient or q-binomial coefficient n _ qn - 1 qn 1 - 1 qn-r 1 - 1 Wq qr - 1 qr 1 - 1 q - 1 is equal to the number of r-dimensional vector subspaces of the n-dimensional vector space Vn q over the finite field GF q . We let Gn Gn q denote the total number of vector subspaces of Vn q . The numbers Gn were named the Galois numbers by Goldman and Rota 4 p. 77 . Goldman and Rota 4 proved the recursion formula Gn 1 2Gn qn - 1 Gn-1 1 for the Galois numbers. Nijenhuis Solow and Wilf 4 gave a different proof of 1 by using the observation that the r-dimensional vector subspaces of Vn q are in one-to-one correspondence with the n by n matrices over GF q which have rank r and are in reduced row echelon form rref . Recall that such a matrix is in rref if its last n - r rows are all zeros in each of the first r rows the first nonzero entry is a 1 the index of the i-th column called a pivotal column in which one of these r 1 s .