Irreducibility results for parabolic induction of representations of the general linear group over a local non-archimedean field can be formulated in terms of Kazhdan--Lusztig polynomials of type $A_n$. Spurred by these results, we hypothesize a simple identity for certain alternating sums of $2^n$ Kazhdan-Lusztig polynomials with respect to $S_{2n}$.
Equivalence between invariance conjectures for parabolic Kazhdan-Lusztig polynomials
An Erdős-Ko-Rado theorem in general linear groups
On combinatorial invariance of parabolic Kazhdan-Lusztig polynomials
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