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10
votes
1answer
209 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
0answers
96 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 ...
34
votes
1answer
896 views

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

In their seminal 1979 paper here, Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the corresponding Iwahori-Hecke algebra. In particular they showed how to pass from a standard ...
0
votes
1answer
91 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 ...
8
votes
3answers
576 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
2answers
370 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
1answer
322 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 ...
8
votes
1answer
328 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 ...
2
votes
0answers
207 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
1answer
378 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 ...