# Birkhoff stratification

As an immediate consequence of Birkhoff factorization, [a1], the group of differentiable invertible matrix loops $\operatorname{LGL} ( n , \mathbf{C} )$ may be decomposed in a union of subsets $B _ { \kappa }$, labelled by unordered $n$-tuples of integers $\kappa$. Each of these consists of all loops with $\kappa$ as the set of partial indices. This decomposition is called a Birkhoff stratification. It reflects important properties of holomorphic vector bundles over the Riemann sphere [a2], singular integral equations [a3], and Riemann–Hilbert problems [a4]. The structure of a Birkhoff stratification resembles those of Schubert decompositions of Grassmannians and Bruhat decompositions of complex Lie groups (cf. also Bruhat decomposition). The Birkhoff strata $B _ { \kappa }$ are complex submanifolds of finite codimension in $\operatorname{LGL} ( n , \mathbf{C} )$. Codimension, homotopy type and cohomological fundamental class of $B _ { \kappa }$ are expressible in terms of the label $\kappa$ [a5]. The adjacencies among the Birkhoff strata describe deformations of holomorphic vector bundles [a5]. Birkhoff stratifications also exist for loop groups of compact Lie groups [a6]. For the group of based loops $\Omega G$ on a compact Lie group $G$, the Birkhoff strata are contractible complex submanifolds labelled by the conjugacy classes of homomorphisms $\mathcal{T} \rightarrow G$ [a6]. Birkhoff stratification has a visual interpretation in the framework of Morse theory of the energy function on $\Omega G$ [a6]. Certain geometric aspects of Birkhoff stratification may be described in terms of non-commutative differential geometry and Fredholm structures [a7], [a8]. In particular, the Birkhoff strata become Fredholm submanifolds of $\Omega G$ endowed with various Fredholm structures. Fredholm structures on $\Omega G$ arise from the natural Kähler structure on $\Omega G$ [a7] and in the context of generalized Riemann–Hilbert problems with coefficients in $G$ [a8]. Curvatures and characteristic classes of Birkhoff strata may be computed in the spirit of non-commutative differential geometry, in terms of regularized traces of appropriate Toeplitz operators [a7].