# Tagged Questions

531 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 ...
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### 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 ...
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### elements in the weyl group

Let W be the Weyl a group of a semisimple simply connected group over C. Let I={1,...,r} the set of simple roots. For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...
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### Are there Hamilton paths in Cayley graphs of Coxeter groups?

Hi everyone. I want to optimize certain computation on finite Coxeter groups $(W,S)$. Basically I compute the matrices $\rho(T_w)$ for all $w\in W$ of a matrix representation $H\to K^{d\times d}$ of ...
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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 ...
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### Spectrum of adjacency matrix of simple Lie algebra.

Let $\mathfrak{g}$ be a finite dimensional, simple, complex Lie algebra. Define the Coxeter adjacency matrix to be the matrix $A=2I-C$ where $C$ is the Cartan matrix of $\mathfrak{g}$. Let ...
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### Presentation of the pure Artin groups

Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m_{ij}$ . The presentation of $W$ is given by W= < T_1, \dots, T_n | T_i^2=1, T_iT_jT_i \ldots = T_jT_iT_j \ldots, i \neq ...
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### Subexpressions of reduced words in Coxeter groups

Let $\underline{w} = [s_1, s_2, \dots ,s_n]$ be a reduced expression in a Coxeter group $W$. Given $x$ in $W$ one can consider the set $\Pi(\underline{w},x)$ consisting of all subexpressions of ...
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### How linearly independent are the obvious combinatorial invariants of a Bruhat interval?

Let $[u, v]$ be a Bruhat interval in some Coxeter group. Let $I$ be the set of all Bruhat intervals. I am interested in functions $I \to \mathbb{Z}$ which are invariant under poset isomorphisms. ...
Let $(W, S)$ be a Coxeter system, and let $T = \bigcup_{w \in W, s \in S} wsw^{-1}$. Associated to every element $t \in T$ is a unique positive root $\alpha_t \in \Phi^{+}$ considered as a vector in ...
Let $(W, S)$ be a Coxeter system. Soergel defined a category of bimodules $B$ over a polynomial ring whose split Grothendieck group is isomorphic to the Hecke algebra $H$ of $W$. Conjecturally, the ...