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There are two types of parabolic Kazhdan Lusztig polynomials, namely, of type -1: $P_{x,w}^{I,-1}$ and of type $q$: $P_{x,w}^{I,q}$. See Kazhdan–Lusztig and R-Polynomials, Young’s Lattice, and Dyck Partitions

My question: What is the meaning of $-1$ and $q$?

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  • $\begingroup$ It woulod be helpful to refer to Deodhar's 1987 paper and Spergel's 1997 paper for the broader context. $\endgroup$
    – Jim Humphreys
    Jun 3, 2019 at 1:45

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These polynomials are connected to the canonical basis of the induction from a parabolic subalgebra $H_I$ up to the whole Hecke algebra $H$. The difference is which module is being induced: The $q$-variant induces the trivial module (on which the generators $T_s$ of $H_I$ act as multiplication by $q$) while the $(-1)$ variant induces the sign module (on which the generators act as multiplication by $-1$).

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  • $\begingroup$ Thank you for your detailed explanation. $\endgroup$
    – James Cheung
    Mar 16, 2019 at 13:57
  • $\begingroup$ 1.I would like to how to induce from $H_I$ to $H$? 2. Is $H_I$ the Hecke algebra of Coxeter system $(W_I,I)$. 3. I would like to know is $H_I$-module, or the parabolic subalgebra $H_I$, or both being induced? $\endgroup$
    – James Cheung
    May 4, 2019 at 7:07
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these are polynomials in $q$ of two types, which satisfy either of the two recursions: $$P_{v,w}^{I,q}=-P_{v,ws}^{I,q}\;\;\text{or}\;\;P_{v,w}^{I,-1}=qP_{v,ws}^{I,-1},$$ see for example these lecture notes.

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  • $\begingroup$ Thank you for your notes. $\endgroup$
    – James Cheung
    Mar 16, 2019 at 13:57
  • $\begingroup$ How did you prove the above recursions, do you have any references of the proof? I cannot figure the proof out and doubt whether they are actually valid. $\endgroup$
    – James Cheung
    Sep 7, 2019 at 6:19

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