Enhanced Bruhat decomposition and Morse theory

Morse function is called strong if all its critical values are pairwise distinct. Given such a function $f$ and a field $\mathbb{F}$ Barannikov constructed a pairing of some of the critical points of $f$, which is now also known as barcode. With every Barannikov pair we naturally associate (up to sign) an element of $\mathbb{F} \setminus \{0\}$; we call it Bruhat number. The paper is devoted to the study of these Bruhat numbers. We investigate several situations where the product of all these numbers (some being raised to the power $-1$) is independent of $f$ and interpret it as a Reidemeister torsion. We apply our results in the setting of one-parameter Morse theory by proving that generic path of functions must satisfy a certain equation mod 2 (this was initially proven in \cite{Akhm} under additional assumptions). On the linear-algebraic level our constructions are served by the following variation of a classical Bruhat decomposition for $GL(\mathbb{F})$. A unitriangular matrix is an upper triangular one with 1's on the diagonal. Consider all rectangular matrices over $\mathbb{F}$ up to left and right multiplication by unitriangular ones. Enhanced Bruhat decomposition describes canonical representative in each equivalence class.