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A Gröbner basis for Kazhdan-Lusztig ideals
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 4, August 2012
- pp. 1089-1137
- 10.1353/ajm.2012.0031
- Article
- Additional Information
{\it Kazhdan-Lusztig ideals}, a family of generalized determinantal ideals
investigated in [Woo-Yong~'08], provide an explicit choice of coordinates
and equations encoding a neighborhood of a torus-fixed point of a Schubert
variety on a type $A$ flag variety. Our main result is a Gr\"{o}bner basis
for these ideals. This provides a single geometric setting to
transparently explain the naturality of pipe dreams on the {\it Rothe
diagram of a permutation}, and their appearance in: \begin{itemize}
\item combinatorial formulas [Fomin-Kirillov '94] for Schubert and
Grothendieck polynomialsof [Lascoux-Sch\"{u}tzenberger '82];
\item the equivariant $K$-theory specialization formula of
[Buch-Rim\'{a}nyi '04]; and
\item a positive combinatorial formula for multiplicities of Schubert
varieties in good cases, including those for which the associated
Kazhdan-Lusztig ideal is homogeneous under the standard grading.
\end{itemize} Our results generalize (with alternate proofs) [Knutson-Miller '05]'s
Gr\"{o}bner basis theorem for Schubert determinantal ideals and their
geometric interpretation of the monomial positivity of Schubert
polynomials. We also complement recent work of [Knutson '08 $\&$ '09] on
degenerations of Kazhdan-Lusztig varieties in general Lie type, as well as
work of [Goldin '01] on equivariant localization and of [Lakshmibai-Weyman
'90], [Rosenthal-Zelevinsky '01], and [Krattenthaler '01] on Grassmannian
multiplicity formulas.