Birkhoff factorization explained

with coefficients that are Laurent polynomials in into a product , where has entries that are polynomials in , is diagonal, and has entries that are polynomials in . There are several variations where the general linear group is replaced by some other reductive algebraic group, due to .

Birkhoff factorization implies the Birkhoff–Grothendieck theorem of that vector bundles over the projective line are sums of line bundles.

Birkhoff factorization follows from the Bruhat decomposition for affine Kac–Moody groups (or loop groups), and conversely the Bruhat decomposition for the affine general linear group follows from Birkhoff factorization together with the Bruhat decomposition for the ordinary general linear group.

See also

• Birkhoff decomposition (disambiguation)
• Riemann–Hilbert problem