Questions tagged [kazhdan-lusztig]

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6 votes
1 answer
102 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 ...
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4 votes
1 answer
222 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{...
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7 votes
0 answers
178 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-...
8 votes
1 answer
229 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^\...
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1 vote
0 answers
107 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 ...
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4 votes
0 answers
58 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?
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3 votes
1 answer
191 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 ...
8 votes
3 answers
497 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: ...
3 votes
2 answers
288 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 ...
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3 votes
0 answers
96 views

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)-\...
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0 votes
1 answer
179 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 ...
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2 votes
0 answers
105 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_{...
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5 votes
0 answers
109 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 ...
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2 votes
0 answers
52 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 ...
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10 votes
2 answers
886 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 ...
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3 votes
1 answer
150 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 ...
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2 votes
0 answers
41 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 ...
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3 votes
0 answers
132 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 ...
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1 vote
0 answers
93 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 ...
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4 votes
1 answer
301 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 ...
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4 votes
2 answers
411 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 ...
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6 votes
1 answer
239 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 \...
1 vote
0 answers
78 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?
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3 votes
0 answers
101 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 $\...
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2 votes
1 answer
95 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 $x,y\in W$?
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4 votes
1 answer
291 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 $q$-...
5 votes
1 answer
175 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 $(...
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1 vote
2 answers
168 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 ...
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6 votes
0 answers
177 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 ...
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4 votes
1 answer
226 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}{...
3 votes
2 answers
270 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 $\ell(sw) < \ell(w)$, ...
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1 vote
1 answer
123 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 ...
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1 vote
0 answers
31 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 ...
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3 votes
0 answers
354 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 ...
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5 votes
0 answers
168 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 $\mathfrak{g}_F=\mathfrak{g}\otimes F$....
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3 votes
1 answer
207 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 ...
  • 1,171
6 votes
0 answers
249 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 ...
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15 votes
1 answer
595 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 \...
4 votes
0 answers
141 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 $s_i$...
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1 vote
1 answer
148 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 ...
9 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 ...
2 votes
2 answers
856 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 ...
3 votes
1 answer
435 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 ...
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 ...
8 votes
1 answer
667 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 ...
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6 votes
1 answer
565 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 ...
3 votes
1 answer
544 views

Are Kazhdan-Lusztig $R$-polynomials the Poincare polynomials of the corresponding affine varieties

Let $x$ and $y$ be two elements of $S_n$. Let $U(x,y)$ be the intersection $(BxB \cap B_{-} y B)/B$ inside the flag variety. Here $B$ and $B_{-}$ are the groups of upper and lower-triangular matrices ...