Bruce Sagan (Michigan State)
GCD Determinants
In 1876, H. J. S. Smith proved the following beautiful determinantal
identity. Let M be an n×n matrix with entries
mi,j=gcd(i,j), where, as usual, gcd stands for the greatest
common divisor. Then
- det M = φ(1) φ(2) ···
φ(n),
where φ is the Euler phi-function, i.e., φ(n) is the number of
positive integers m less than or equal to n with gcd(m,n)=1. Since Smith's
paper, a host of generalizations and analogues have appeared in the
literature. I will show that many of them are special cases of a simple
identity in the incidence algebra of an arbitrary poset P. Smith's
original result then follows by Möbius inversion when P is the
lattice of divisors of n.
This is joint work with E. Altinisik and N. Tuglu.