Editing Character table (section)

=== Calculating the irreducible representation Γ<sub>irreducible</sub> from the reducible representation Γ<sub>reducible</sub> along with the character table ===
From the above discussion, a new character table for a water molecule (<math>C_{2v}</math> point group) can be written as
: {| class="wikitable"
|+New character table for <chem>H2O</chem> molecule including <math>\Gamma_{\text{red}}</math>
!
!<math>E</math>
!<math>C_2</math>
!<math>\sigma_{v(xz)}</math>
!<math>\sigma'_{v(yz)}</math>
|-
|<math>A_1</math>
|1
|1
|1
|1
|-
|<math>A_2</math>
|1
|1
| −1
| −1
|-
|<math>B_1</math>
|1
| −1
|1
| −1
|-
|<math>B_2</math>
|1
| −1
| −1
|1
|-
|<math>\Gamma_{\text{red}}</math>
|9
| −1
|3
|1
|}
Using the new character table including <math>\Gamma_{\text{red}}</math>, the reducible representation for all motion of the <chem>H2O</chem> molecule can be reduced using below formula

: <math>N = \frac{1}{h}\sum_{x}(X^x_i \times X^x_r\times n^x)</math> 

where, 

: <math>h =</math> order of the group, 
: <math>X^x_i =</math> character of the <math>\Gamma_{\text{reducible}}</math> for a particular class, 
: <math>X^x_r =</math> character from the reducible representation for a particular class, 
: <math>n^x =</math> the number of operations in the class

So,

<math>N_{A_1} = \frac{1}{4}[(9\times 1\times 1)+((-1)\times 1\times 1)+(3\times 1\times 1)+(1\times 1\times 1)] = 3</math>

<math>N_{A_2} = \frac{1}{4}[(9\times 1\times 1+((-1)\times 1\times 1)+(3\times(-1)\times 1)+(1\times(-1)\times 1)] = 1</math>

<math>N_{B_1} = \frac{1}{4}[(9\times 1\times 1)+((-1)\times(-1)\times 1)+(3\times 1\times 1)+(1\times(-1)\times 1)] = 3</math>

<math>N_{B_2} = \frac{1}{4}[(9\times 1\times 1)+((-1)\times(-1)\times 1)+(3\times(-1)\times 1)+(1\times 1\times 1)] = 2</math>

So, the reduced representation for all motions of water molecule will be

<math>\Gamma_{\text{irreducible}} = 3A_1 + A_2 + 3B_1 + 2B_2</math>