4
$\begingroup$

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 $q$?

$\endgroup$
6
$\begingroup$

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 $-1$).

$\endgroup$
  • $\begingroup$ Thank you for your detailed explanation. $\endgroup$ – James Cheung Mar 16 at 13:57
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Thank you for your notes. $\endgroup$ – James Cheung Mar 16 at 13:57

Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.