Monteith Building

In a typical undergraduate linear algebra course, perhaps the main computational technique introduced for solving systems of linear equations is that of Gaussian elimination, a process that takes any matrix and converts it into its reduced row echelon form (RREF). One obtains the RREF of a matrix \(M\) by performing a sequence of invertible elementary row operations in such a way that row \(i\) of RREF(\(M\)) only depends on rows \(1,2,\dotsc,i\) of \(M\); equivalently, one multiplies \(M\) on the left by a particular invertible lower triangular matrix. In this way, the orbits of the left action of the group of invertible lower triangular \(k\) by \(k\) matrices on the space of \(k\) by \(n\) matrices are in natural bijection with the set of \(k\) by \(n\) matrices in reduced row echelon form. One can then take a coarser decomposition than this by looking at the subspaces of matrices whose RREFs share the same pivot entries and study this decomposition through the lens of algebraic geometry. This basic theme has an endless number of interesting variations, which collectively form one entry point to the very rich combinatorial and algebro-geometric theory of Schubert calculus and the Bruhat decomposition. Over the course of this Halloween algebraic combinatorics escapade, we will first define and examine the classical Schubert calculus and Bruhat decomposition in the general linear group \(\mathrm{GL}_n(C)\) and the corresponding flag variety \(\mathrm{Fl}_n\). Our investigation will then naturally lead us to one of the many equivalent definitions of the Bruhat order on the symmetric group \(S_n\), a poset structure with fascinating combinatorics and a powerful tool for analyzing the relevant geometry. From here, we will take a scenic detour into the land of Coxeter systems and diagrams, aspiring toward some purely combinatorial descriptions of the Bruhat order on \(S_n\). Lastly, we will reinterpret the general theory of Bruhat orders for finite Coxeter systems in a geometric dialect, discussing how the Bruhat decomposition manifests in reductive complex algebraic groups \(G\) beyond the type A case of \(G=\mathrm{GL}_n\).

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