Complexity of calculating the determinant

Question

For calculating the determinant, there are several algorithms available:

• Laplace: $\mathcal{O}{(n!)}$
• Bareiss: $\mathcal{O}{(n^3)}$
• LU-decomposition: $\mathcal{O}{(n^3)}$
• Strassen: $\mathcal{O}{(n^{2.807})}$
• Coppersmith-Winograd: $\mathcal{O}{(n^{2.376})}$

Now my question is: What is the theoretical limit for calculating the determinant. It is $\mathcal{O}{(n^2)}$, isn't it? If yes, how is this possible?

Is this still an active research field, or is the Coppersmith-Winograd-Algorithm "the limit"?

Answer 1

That algorithm's Wikipedia page mentions slight improvements for matrix multiplication, which has the same complexity as determinant calculation. The best is due to Le Gall 2014, reducing the exponent to $2.3728639$.