# Questions tagged [kazhdan-lusztig]

The kazhdan-lusztig tag has no usage guidance.

51
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### Comparison of IC sheaves on Schubert varieties on two settings (l-adic vs. complex)

This question is basically about comparison of IC sheaves (or their sheaf cohomologies) for the settings: 1. variety is over $\mathbb{C}$ and sheaf is $\mathbb{C}$-linear, 2. variety is over a finite ...

6
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1
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### An alternative form of the Kazhdan-Lusztig conjecture

Fix a complex semisimple Lie algebra $\mathfrak{g}$. Denote by $W$ the corresponding Weyl group, with length function $\ell$ and Bruhat order $\leq$. Let $\lambda$ be an integral anti-dominant weight. ...

8
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### What is the remaining difficulty in the proof of the Humphreys conjecture (on the support variety of tilting modules)?

Humphreys conjecture describes the support variety of tilting modules using the correspondence between two-sided cells and nilpotent orbits. But the support variety can also be given by Lusztig–Vogan ...

3
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### Evaluations of Kazhdan-Lusztig polynomials at $q=-1$ and $q=1$

I am looking for some references about the meaning (if any) of the evaluations of Kazhdan-Lusztig polynomials of Coxeter groups at the integers $1$ and $-1$. Even if this is meaningful only by ...

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### Are standard Koszul algebras with the same Kazhdan-Lusztig polynomials Morita equivalent?

Let $A,B$ be a pair of quasi-hereditary algebras and assume that $A$ and $B$ are both standard Koszul. Further assume that the graded decomposition matrices of $A$ and $B$ coincide (that is, the ...

4
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1
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### A problem on Kazhdan–Lusztig theorem

I am reading Chriss, Ginzburg's book Representation theory and complex geometry. In theorem 7.2.16 it says that the convolution action of the Steinberg variety $St=\tilde{\mathcal{N}}\times_\mathfrak{...

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### A reference for Bernstein's approach to KL conjectures

The exposition in the famous J. Bernstein's lecture notes on D-modules is rather sketchy.
Fortunately, there are many other sources for the information from this text (like Hotta's excellent "D-...

8
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251
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### Branching rule for Specht modules over Kazhdan-Lusztig basis

Suppose $\lambda\vdash n$ is a partition and $S^\lambda$ is the associated irreducible representation of $S_n$. As we know from the branching rule, we have an isomorphism of $S_{n-1}$ modules
$$S^\...

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### Kazhdan-Lusztig Conjecture over non-algebraically closed field

Let $G$ be a split connected semi-simple (or reductive) algebraic group over a (non-archimedean) field $k$ of characteristic zero. Denote by $\mathfrak{g}=\mathrm{Lie}(G)$ the semi-simple (or ...

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### Parabolic Bruhat graphs for exceptional types

I am looking for some computer software or a reference for some parabolic Bruhat graphs. In particular, what I really need $E_8 \setminus E_7$. Does anyone know where or how I'd find this?

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### Description of Soergel modules

So this is asking a basic and/or stupid question (my apology and appreciation) about Soergel modules that comes out of exercises by me who knows little about the subject.
Let $W$ be a finite Weyl ...

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### Motivation for the Kazhdan-Lusztig involution

I would like to know about the motivation behind the Kazhdan–Lusztig involution on an Iwahori–Hecke algebra.
I'll borrow the conventions from Libedinsky's Gentle introduction to Soergel bimodules I: ...

3
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2
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339
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### Morphism of Verma modules

$\DeclareMathOperator\Hom{Hom}$I'm trying to understand morphism of Verma modules and consider the following example.
PART 1:
Consider $\mathfrak{g}=\mathfrak{gl}_3$ over $\mathbb{C}$ with positive ...

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### Kazhdan-Lusztig polynomials and the defect of a Bruhat interval

Let $(W,S)$ be a Coxeter system with length function $\ell$ and $T=\bigcup_{w\in W}wSw^{-1}$.
Set
$N(u,v):=\{t\in T: u< tu \le v\}$,
$\overline{\ell}(u,v):=|N(u,v)|$,
$\ell(u,v):=\ell(v)-\...

0
votes

1
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203
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### When does the Kazhdan-Lusztig polynomial $P_{x,w}(q)$ not vanish at $q=1$?

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra. For any $\lambda\in \mathfrak{h}^{*}$ let $M(\lambda)$ and $L(\lambda)$ be the Verma module and the simple ...

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### Monotonicity Theorem of inverse Kazhdan Lusztig polynomials

Let $P_{x,w}$ and $Q_{x,w}$ be the Kazhdan Lusztig polynomial and the inverse Kazhdan Lusztig polynomial of Coxeter group $W$, respectively. i.e., $\sum_{x\le y\le z}(-1)^{\ell(y)-\ell(x)}P_{x,y}(q)Q_{...

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### Progress on the result about montonicity of Kazhdan Lustzig polynomials

I am reading the paper Masato Kobayashi---Combinatorics on Bruhat Graphs and
Kazhdan-Lusztig Polynomials.
Let $P_{x,w}$ be the Kazhdan Lusztig polynomial of $W$.
There is a result about ...

2
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### Criterion for parabolic Kazhdan-Lusztig polynomials to be monic power of $q$

Let $(W,S)$ be a Coxeter system, $I\subseteq S$, $W^I=\{w\in W: sw>w,\ \forall s\in I\}$, $P^{I,q}_{x,w}(q)$ be the parabolic Kazhdan-Lusztig polynomial of $W^I$ of type $q$.
In KAZHDAN–LUSZTIG ...

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### Strange formulas that gave rise to Koszul duality

According to p.8 of the note KOSZUL DUALITY AND APPLICATIONS IN
REPRESENTATION THEORY by Geordie Williamson.
Let $M(\eta)$ be the Verma module of weight $\eta$, $L(\eta)$ be its unique simple ...

3
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1
answer

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### Bruhat ordering and non-vanishing Extension groups

Let $P_{x,w}(q)$ be the Kazhdan Lusztig polynomial. It is well-known that $P_{x,w}(q)\neq 0\iff x\le w$.
By the interpretation of the Kazhdan Lusztig polynomial in terms of extension group, it holds ...

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### About monotonicity of parabolic Kazhdan Lusztig polynomials

Let $W_I$ be the parabolic subgroup generated by $I$, ${}^IW$ be the set of minimal length right coset representative of $W_I$ in $W$, and $w_I$ be the longest element in $W_I$.
Let $P_{u,v}$ be the ...

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### Doubt concerning certain Extension groups

I am reading the paper "Categories Of Highest Weight Modules: Applications To Classical Hermitian Symmetric Pairs"
I do NOT understand why Proposition 14.4 implies Theorem 14.9.
Now let us use the ...

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### About parabolic Kazhdan-Lusztig polynomials

Denote by $P^{I,y}_{x,w}$ be the parabolic Kazhdan-Lusztig polynomial of ${}^IW$ of type $y$.
I have heard that the polynomials $P^{I,q}_{x,w}$ give the transition matrix between a canonical basis ...

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### Kazhdan–Lusztig polynomials in terms of Ext groups

Let $P_{x,w}$ be the Kazhdan–Lusztig polynomial, $\rho$ be the half sum of positive roots in $\Phi^+$, $M_x$ be the Verma module with highest weight $x\cdot(-2\rho)$ and $L_w$ be the simple highest ...

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### About parabolic Kazhdan Lusztig polynomials

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 ...

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### Schutzenberger's evacuation and $\mu$-coefficient of Kazhdan–Lusztig polynomials

$\def\SYT{\mathrm{SYT}}\def\RSK{\mathrm{RSK}}\DeclareMathOperator\evac{evac}$Let $\mathfrak{S}_n$ be the symmetric group, $\SYT_n$ be the set of standard young tableaux of size $n$.
For $u\in \...

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### About Kazhdan Lusztig polynomial evaluating at q=1

Given $w\le w'$ (in Bruhat ordering), does $P_{x,w}(1)\le P_{x,w'}(1)$ (in usual ordering of $\mathbb{R}$), where $P_{x,w}(q)$ is the Kazhdan Lusztig polynomial?

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### Relationship bewteen Kazhdan-Lusztig Vogan polynomial and Kazhdan-Lusztig polynomial

Let $W^I=\{w\in W: w^{-1}\Phi_I^+\subseteq \Phi^+\}$, $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$
Let $M(\lambda)$ be the Verma module with highest weight $\...

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1
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### About Kazhdan-Lusztig polynomial

Let $(W,S)$ be a Coxeter system. One can have the Kazhdan-Lusztig polynomial $P_{x,\ y}(q)$.
Does $P_{x,\ y}(q)=P_{x^{-1},\ y^{-1}}(q)$ for all $x,y\in W$?

4
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### Examples of non-trivial Kazhdan-Lusztig polynomials

I'm looking for examples of non-trivial Kazhdan-Lusztig polynomials, specifically in the case where the Coxeter system is a Weyl group.
For example, the simplest polynomial with non-trivial $q$-...

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1
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### Parabolic Kazhdan-Lusztig polynomial coincide?

Let $(W,S)$ be a Coxeter system. For any subset $I\subseteq S$, we can have the parabolic Kazhdan-Lusztig polynomial $P_{x,w}^I(q)$ with respect to $I$.
Now consider $I\subseteq J\subseteq S$. Both $(...

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2
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### About Morita equivalent and regular block of category $\mathcal{O}^\mathfrak{p}$

In the following paper: Representation type of the blocks of category $\mathcal{O}_S$
https://www.sciencedirect.com/science/article/pii/S0001870804002853
On p.196, it states that "When $\mu$ is ...

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### Combinatorics of $p$-Kazhdan--lusztig polynomials

When can we (and can we not!) understand the dimensions of simple modules, $D(\lambda)$, of symmetric groups in a combinatorial fashion?
Let's assume that I'm going to try to do this using the theory ...

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### Number of pairs of permutation in $S_n$ whose $\mu$-coefficient (of their Kazhdan Lusztig polynomial) is non-zero

I am interested in how many pairs of permutations $(u,w)$ in $S_n$, such that the $\mu$-coefficient of its Kazhdan-Lusztig polynomial $P_{u,w}(q)$ is non-zero? where $\mu_{u,w}=[q^{\frac{l(w)-l(u)-1}{...

3
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2
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### Recursive formula for inverse Kazhdan-Lusztig polynomials

Let $(W,S)$ be a Coxeter group. Then the Kazhdan-Lusztig polynomials $P_{x,w}$ are known to satisfy the following recursive identity:
For $x < w$ and $s \in S$ satisfying $\ell(sw) < \ell(w)$, ...

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1
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### About left cell of a permutation

I am reading a paper Cellular algebras by J.J. Graham, G.I. Lehrer. I do not understand the follwing words labelled by yellow.
First, I know Robinson-Schensted correspondence of a permutation in the ...

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### Cellular basis of $KW(B_2)$

Take $K$ a field. Let $W(B_2)$ be the coxeter group of type $B_2$ with a set $\{s_0,s_1\}$ of generators. Then $W(B_2)=\{1, s_0,s_1,s_0s_1,s_1s_0,s_0s_1s_0, s_1s_0s_1,s_1s_0s_1s_0\}$. I know that the ...

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### Kazhdan-Lusztig basis of the symmetric group algebra $\mathbb{K}S_n$

Let $S_n$ be the symmetric group on the set $\{1,2,\ldots,n\}$ and $\mathbb{K}$ a field. Denote by $\mathbb{K}S_n$ the symmetric group algebra.
What's Kazhdan-Lusztig basis of $\mathbb{K}S_n$? I ...

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### From parahorics to conjugacy classes in the Weyl group: a question about the Kazhdan-Lusztig map

Let $\mathfrak{g}$ be a simple Lie algebra. Let $F=\mathbb{C}((t))$ and $A=\mathbb{C}[[t]]$. Let $\mathfrak{g}_A:=\mathfrak{g}\otimes_\mathbb{C} A$ and similarly $\mathfrak{g}_F=\mathfrak{g}\otimes F$....

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### Is Koszulity equivalent to the Lusztig character formula holding?

Let $\pi$ denote a saturated set of weights. Let $S_q(\pi)$ denote the associated generalised $q$-Schur algebra. I was wondering if the following claim is true:
Claim: The algebra $S_q(\pi)$ is ...

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### Papers/Programs for computing periodic KL polynomials?

Periodic Kazhdan-Lusztig polynomials (for an affine Weyl group) are polynomials that control Jordan-Holder multiplicities for certain representations ("baby Verma modules") of an algebraic group in ...

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### Subquotients in the Verma filtration on Verma modules

Let $\lambda$ be a dominant integral weight of $\mathfrak g$, a finite-dimensional reductive Lie algebra over $\mathbb C$. Let $M(w\cdot \lambda)$ denote the Verma module with high weight $w\cdot \...

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### Degeneration of modules over the affine symmetric group and jeu de taquin

Let $H_n$ be the group algebra of the affine Coxeter group of type A (feel free to replace it by the affine Hecke algebra). This is generated by elements $y_i$'s, $i=1,\dots,n$ and transpositions $s_i$...

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### Decomposition of C' Kazhdan-Lusztig basis element associated to longest word in S_n

I'm trying to decompose the Kazhdan-Lusztig C' basis element associated to the longest word in $S_n$, $C'_{w_0}$ into products and sums of elements $C'_w$ where $w < w_0$ in the Bruhat order. For ...

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3
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### Is there a list of Kazhdan-lusztig polynomials?

When studying the combinatorics and representation of Coxeter groups, I often find it irksome to compute KL- and R- polynomials from scratch on Maple or Sage. The time it consumes to generate a ...

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2
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### Around the socle filtration of a Verma module

Work in the context of a finite dimensional simple Lie algebra over $\mathbb{C}$.
Write $W$ for the Weyl group and $\leq$ for the Bruhat order. For $w\in W$ let $\Delta_w$ denote the Verma module of ...

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### A cohomology computation request.

The short: Let
$X= \{(x,y,z) \in \mathbb{C}^* \times \mathbb{C} \times \mathbb{C} \, |\, yz-x\neq 0\}$
Compute $H^*_c(X)$ (say with $\mathbb{C}$-coefficients).
The long: Unless I messed something ...

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### Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials?

In their seminal 1979 paper Representations of Coxeter groups and Hecke algebras (Invent. Math. 53, doi:10.1007/BF01390031),
Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the ...

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### Schubert varieties which admit small resolutions of singularities

I am looking for an (incomplete) list of partial flag varieties for
which all Schubert cells admit small resolutions of singularities.
This is interesting, for many reasons. My motivation is, that a ...

6
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1
answer

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### When are parabolic Kazhdan-Lusztig polynomials nonzero?

Let $W$ be a Coxeter group with simple reflections $S$ and let $J \subseteq S$. Let $P^J_{\tau, \sigma}$ be the parabolic Kazhdan-Lusztig polynomials in the case $u = q$ in the sense of On Some ...