A Gröbner basis for Kazhdan-Lusztig ideals (Q2903091)

Let \(z=\{z_{ij}\}_{1\leq i,j\leq n}\) be variables and \(v,w\in S_n\) permutations. Let \(Z^{(v)}\) be the specialized generic matrix, i.e. with entries \(z_{ij}\) but specialized to \(z_{n-v(i)+1,i}=1\), \(z_{n-v(i)+1,a}=0\;,\;z_ {b,i}=0\) if \(a>i, b>n-v(i)+1\). Let \(z^{(v)}\subset z\) be the remaining variables. Let \(r^w_{i,j}=\#\{k\;|\;w(k)\geq n-i+1, k\leq j\}\) and denote by \(Z^{(v)}_{ab}\) the southwest \(a\times b\) submatrix of \(Z^{(v)}\). The Kazhdan--Lusztig ideal \(I_{v,w}\subseteq\mathbb{C}[z^{(v)}]\) is the ideal generated by all minors of size \(1+r^w_{ij}\) of \(Z^{(v)}_{ij}\) for all \(i,j\) (the so--called defining minors). Consider the following monomial ordering: NEWLINE\[NEWLINE z_{ij}<z_{kl}, \text{ if } j<l\;\text{ or if } j=l\text{ and } i<k\;. NEWLINE\]NEWLINE It is proved that the defining minors are a Gröbner basis of \(I_{v,w}\).NEWLINENEWLINEKazhdan--Lusztig ideals provide an explicite choice of coordinates and equations encoding a neighborhood of a torus--fixed point of a Schubert variety on a type \(A\) flag variety.NEWLINENEWLINEThe Gröbner basis turns out to be a tool to explain several combinatorical formulas.