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56 R. GOW AND T. J. LAFFEY
Since Q is a permutation matrix, we see that the Smith invariants of Q(x) are the monomials xw(i,j) running over those (i, j) with qij = 1. This completes the proof.
Example1.2. Form=3,n=5
P5,3=5 1 0.
In characteristic 2,
where
0 1 1
P5,3 =1 1 0=LQU,
100
1 0 1
011 001 001
Hence the Jordan canonical form of J3 ⊗ J5 in characteristic 2 is J4 ⊕J4 ⊕J7.
10 5 1 100
0 1 0 Positions (1,2) and (2,1) have weight 4, while position (3,3) has weight 7.
1 0 0
L=1 1 0, U=0 1 1,andQ=1 0 0.
In characteristic 3,
where
1 2 1
P5,3 =2 1 0=LQU,
100
1 −1 1 1 0 0 101 001 010
Hence, in characteristic 3, J3 ⊗ J5 has Jordan canonical form J3 ⊕J6 ⊕J6.
In characteristic 5,
0 0 1
P5,3 =0 1 0=Q,
100 so J3 ⊗J5 has Jordan form J5 ⊕J5 ⊕J5.
1 0 0
L=−1 1 0, U=0 1 −1,and Q=0 0 1.
