The authors provide path-based sufficient conditions for a Kazhdan-Lusztig ideal to be inhomogeneous, plus explicit examples outside Neye's standard-homogeneous class.
Governing singularities of schubert varieties
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On the homogeneity problem of the Kazhdan-Lusztig ideals
The authors provide path-based sufficient conditions for a Kazhdan-Lusztig ideal to be inhomogeneous, plus explicit examples outside Neye's standard-homogeneous class.