Questions tagged [kazhdan-lusztig]

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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 ...
Ji Woong Park's user avatar
6 votes
1 answer
360 views

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. ...
Estwald's user avatar
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8 votes
1 answer
310 views

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 ...
211's user avatar
  • 93
3 votes
0 answers
152 views

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 ...
Luis Ferroni's user avatar
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6 votes
1 answer
120 views

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 ...
Chris Bowman's user avatar
  • 1,191
4 votes
1 answer
245 views

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{...
Yang's user avatar
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7 votes
0 answers
196 views

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-...
Rgdn Dznrbx's user avatar
8 votes
1 answer
251 views

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^\...
GossipM's user avatar
  • 221
1 vote
0 answers
113 views

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 ...
KKD's user avatar
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4 votes
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75 views

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?
Chris Bowman's user avatar
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3 votes
1 answer
242 views

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 ...
Cheng-Chiang Tsai's user avatar
8 votes
3 answers
593 views

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: ...
Richard Hepworth's user avatar
3 votes
2 answers
339 views

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 ...
KKD's user avatar
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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)-\...
James Cheung's user avatar
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0 votes
1 answer
203 views

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 ...
Zhaoting Wei's user avatar
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2 votes
0 answers
114 views

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_{...
James Cheung's user avatar
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5 votes
0 answers
113 views

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 ...
James Cheung's user avatar
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2 votes
0 answers
55 views

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 ...
James Cheung's user avatar
  • 1,855
10 votes
2 answers
905 views

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 ...
James Cheung's user avatar
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3 votes
1 answer
159 views

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 ...
James Cheung's user avatar
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2 votes
0 answers
45 views

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 ...
James Cheung's user avatar
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3 votes
0 answers
133 views

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 ...
James Cheung's user avatar
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1 vote
0 answers
99 views

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 ...
James Cheung's user avatar
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4 votes
1 answer
341 views

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 ...
James Cheung's user avatar
  • 1,855
4 votes
2 answers
465 views

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 ...
James Cheung's user avatar
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6 votes
1 answer
270 views

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 \...
Sylvester W. Zhang's user avatar
1 vote
0 answers
82 views

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?
James Cheung's user avatar
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3 votes
0 answers
110 views

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 $\...
James Cheung's user avatar
  • 1,855
2 votes
1 answer
102 views

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 $x,y\in W$?
James Cheung's user avatar
  • 1,855
4 votes
1 answer
333 views

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 $q$-...
Ben McDonnell's user avatar
5 votes
1 answer
183 views

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 $(...
James Cheung's user avatar
  • 1,855
1 vote
2 answers
170 views

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 ...
James Cheung's user avatar
  • 1,855
6 votes
0 answers
195 views

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 ...
Chris Bowman's user avatar
  • 1,191
4 votes
1 answer
235 views

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}{...
Sylvester W. Zhang's user avatar
3 votes
2 answers
299 views

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 $\ell(sw) < \ell(w)$, ...
Seth's user avatar
  • 185
1 vote
1 answer
130 views

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 ...
bing's user avatar
  • 331
1 vote
0 answers
32 views

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 ...
bing's user avatar
  • 331
3 votes
0 answers
387 views

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 ...
bing's user avatar
  • 331
5 votes
0 answers
186 views

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 $\mathfrak{g}_F=\mathfrak{g}\otimes F$....
Dr. Evil's user avatar
  • 2,681
3 votes
1 answer
216 views

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 ...
Chris Bowman's user avatar
  • 1,191
6 votes
0 answers
265 views

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 ...
dhy's user avatar
  • 5,888
16 votes
1 answer
642 views

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 \...
Allen Knutson's user avatar
4 votes
0 answers
151 views

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 $s_i$...
Adrien's user avatar
  • 8,234
1 vote
1 answer
156 views

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 ...
Michael Abel's user avatar
11 votes
3 answers
1k views

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 ...
Harry Huang's user avatar
2 votes
2 answers
929 views

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 ...
Reladenine Vakalwe's user avatar
3 votes
1 answer
472 views

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 ...
Reladenine Vakalwe's user avatar
39 votes
2 answers
2k views

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 ...
Jim Humphreys's user avatar
8 votes
1 answer
752 views

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 ...
Jan Weidner's user avatar
  • 12.9k
6 votes
1 answer
578 views

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 ...
Jonah Blasiak's user avatar