369] 
495 
369. 
ON A PROPERTY OF COMMUTANTS. 
[From the Philosophical Magazine, voi. xxx. (1865), pp. 411—413.] 
I call to mind the definition of a commutant, viz. if in the symbol 
- t 
1 i i («) 
2 2 2 
_PP P 
we permute independently in every possible manner the numbers 1, 2,...p of each of 
the 6 columns except the column marked giving to each permutation its proper 
sign, + or —, according as the number of inversions is even or odd, thus 
±s ±t • •• A x Si (| „ (o) 
2 s 2 t 2 
P t p 
which is to be read as meaning 
¿S Ì t ••• -^1 Sj • • A p Sp t p .., 
the sum of all the (1.2.3 ... p) e ~ x terms so obtained is the commutant denoted by 
the above-mentioned symbol. In the particular case 6 = 2, the commutant is of course 
a determinant : in this case, and generally if 6 be even, it is immaterial which of 
the columns is left unpermuted, so that the (i*) instead of being placed over any 
column may be placed on the left hand of the A ; but when 6 is odd, the function 
has different values according as one or another column is left unpermuted, and the 
position of the (*f*) is therefore material. It may be added that if all the columns 
are permuted, then, if 6 be even, the sum is 1.2... p into the commutant obtained 
by leaving any one column unpermuted ; but if 6 is odd, then the sum is = 0.