The Smith Normal Form of a Matrix Associated with Young’s Lattice

Let R be a commutative ring with 1 and M an m×m matrix over R. We say that M has a Smith normal form (SNF) over R if there exist matrices P,Q ∈ SL(m,R) (so detP = detQ = 1) such that PMQ is a diagonal matrix diag(d1, d2, . . . , dm) with di | di+1 (1 ≤ i ≤ m − 1) in R. It is wellknown that if R is an integral domain and if the SNF of M exists, then it is unique up to multiplication of each di by a unit ui (with u1 · · · um = 1). We see that Smith normal form is a refinement of the determinant, since detM = d1d2 · · · dm. In the case that R is a principal ideal domain (PID), it is well-known that all matrices in R have a Smith normal form. Not very much is known in general. In this work we are interested in the ring Z[x] of integer polynomials in one variable. Since Z[x] is not a PID, a matrix over this ring need not have