I met a problem when reading the book of Novikov et al. "Modern geometry, vol 1".
In page 200 of the book (GTM 93, English version), they say for 2-form $F$, $$ (F\wedge F)_{0123}=-{1\over 2}\epsilon^{ijkl}F_{ij}F_{kl} $$
However I cannot get the coefficients or signs...
The following is my calculation:
Using the definition of coefficient of differential form: $$ F=\sum_{i<j}F_{ij} \mathrm{d}x^i \wedge \mathrm{d}x^j. $$ Hence \begin{align} F\wedge F&=(\sum_{i<j}F_{ij} \mathrm{d}x^i \wedge \mathrm{d}x^j)\wedge (\sum_{k<l}F_{kl} \mathrm{d}x^k \wedge \mathrm{d}x^l)\\ &=\sum_{i<j}\sum_{k<l}F_{ij}F_{kl} \mathrm{d}x^i \wedge \mathrm{d}x^j\wedge \mathrm{d}x^k \wedge \mathrm{d}x^l\\ &= \sum_{i<j}\sum_{k<l}F_{ij}F_{kl}\epsilon^{ijkl} \mathrm{d}x^0 \wedge \mathrm{d}x^1\wedge \mathrm{d}x^2 \wedge \mathrm{d}x^3\\& =\frac{1}{4}F_{ij}F_{kl}\epsilon^{ijkl} \mathrm{d}x^0 \wedge \mathrm{d}x^1\wedge \mathrm{d}x^2 \wedge \mathrm{d}x^3 \end{align}
Again by definition of differential form, $$ F\wedge F=(F\wedge F)_{0123}\mathrm{d}x^0 \wedge \mathrm{d}x^1\wedge \mathrm{d}x^2 \wedge \mathrm{d}x^3. $$
This implies $$ (F\wedge F)_{0123}=\frac{1}{4}F_{ij}F_{kl}\epsilon^{ijkl}. $$
I am not sure where is the problem. The book usually uses calculation with $sgn(\sigma)$, while I use the representation under basis here.
I get even more confused using formulas with $sgn(\sigma)$...
Thanks a lot for your help!
(By the way, as a student majoring in Applied mathematics, I really like these Russian books, with wonderful explanations of concrete calculation and applications in physics and mechanics!)