Questions tagged [kazhdan-lusztig]
The kazhdan-lusztig tag has no usage guidance.
20
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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-...
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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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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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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 ...
5
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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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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?
4
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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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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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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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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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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 $\...
3
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385
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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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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 ...
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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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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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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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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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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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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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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 ...