Determinants and Their Applications in Mathematical Physics

5.4 The Cofactors of the Matsuno Determinant 195

=−(ci−cj)E

ij

Equation (5.4.15) can be proved in a similar manner by appling (5.4.7) and

the equation in itemm= 2. The proof of (5.4.16) is a little more difficult.

Modify (5.4.12) by making the following changes in the parameters. First

i→k, then (r, s)→(i, j), and, finally,k→r. The result is

∑

i,j

†

E

rj

E

ir

=

∑

i

E

ri

E

ir

. (5.4.19)

Now sum (5.4.10) overi, jand refer to (5.4.19) and (5.4.6):

∑

i,j

(ci−cj)E

ij

=

∑

i,j,r,s

E

is

E

rj

−

∑

r





∑

i,j

E

ir

E

rj





=





∑

i,s

E

is









∑

r,j

E

rj



−

∑

r

∑

i

E

ri

E

ir

=

∑

i

E

ii

∑

r

E

rr

−

∑

i,r

E

ri

E

ir

=

∑

i,r

∣

∣

∣

∣

E

ii

E

ir

E

ri

E

rr

∣

∣

∣

∣

=

∑

i,r

E

ir,ir

,

which is equivalent to (5.4.16). The symbol†can be attached to the sum

on the left without affecting its value. Hence, this identity together with

(5.4.7) yields (5.4.17), which can then be expressed in the symmetric form

(5.4.18) in whichr<s.

5.4.4 Third and Fourth Cofactors

The following identities contain third and fourth cofactors ofE:

∑

r,s

(cr−cs)E

rt,st

=

∑

r,s

E

rst,rst

, (5.4.20)

∑

r,s

(cr−cs)E

rtu,stu

=

∑

r,s

E

rstu,rstu

, (5.4.21)

∑

r,s

(c

2

r−c

2

s)E

rs

=2

∑

r,s

crE

rs,rs

, (5.4.22)

∑

r,s

(cr−cs)

2

E

rs

=

∑

r,s,t

E

rst,rst

, (5.4.23)

∑

r,s

(cr−cs)

2

E

ru,su

=

∑

r,s,t

E

rstu,rstu

, (5.4.24)