Why is the Nil-Hecke Algebra appearing? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T02:38:25Z http://mathoverflow.net/feeds/question/49364 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49364/why-is-the-nil-hecke-algebra-appearing Why is the Nil-Hecke Algebra appearing? Peter McNamara 2010-12-14T07:43:16Z 2010-12-14T07:43:16Z <p>The Nil-Hecke algebra is defined to be the subalgebra of the endomorphism ring of $\mathbb{C}[x_1,\ldots,x_n]$ generated by the operators of multiplication by $x_i$ and the divided difference operators $\partial_i$ which act by $f\mapsto \frac{f-s_if}{x_i-x_{i+1}}$ with $s_i$ being the simple reflection that swaps $x_i$ and $x_{i+1}$. I know of two places where this appears.</p> <p>Firstly, in the construction of Schubert polynomials (and hence implicitly in the calculation of the (ordinary or torus equivariant) cohomology of the flag variety).</p> <p>And secondly, it is equal to the convolution algebra $H^{GL_n}_*(X\times X)$ where $X$ is the variety of complete flags in $\mathbb{C}^n$ and we are taking equivariant Borel-Moore homology.</p> <p>Due to the similarity of the objects appearing in each case, I feel as though this can't be a coincidence, but don't see for myself a direct relationship or know where this is written down. This may be because I've picked up both objects "on the fly" without a good understanding of the literature. Anyway the question is, what is the relation between these incarnations of the nil-Hecke algebra that I am telling myself should exist?</p>