User michael zhong - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:03:24Z http://mathoverflow.net/feeds/user/29921 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116716/why-is-there-a-unique-increasing-maximal-path-in-any-bruhat-interval-under-any-re Why is there a unique increasing maximal path in any Bruhat interval under any reflection order? Michael Zhong 2012-12-18T15:35:40Z 2013-01-03T14:34:48Z <p>According to the increaing-Bruhat-path-explanation of Kazhdan-Lusztig $R$-polynomials(in fact $\tilde{R}$-polynomials), and the fact that $[q^{l(x,y)}]\tilde{R}_{x,y}(q)=1$, there is a unique increasing maximal path in any Bruhat interval under any reflection order. I wonder if there is any direct explanation.<br> Thank Professor Woo for suggesting providing more details. So the following is an detailed description of the increaing-Bruhat-path-explanation of Kazhdan-Lusztig $R$-polynomials, which is from Bjorner and Brenti's book `Combinatorics of Coxeter Groups', GTM 231, pp.136-144.<br> Suppose $(W,S)$ is a Coxeter system with $\Phi$ its root system. A total ordering $0$ such that $\lambda\alpha+\mu\beta\in \Phi^+$, then $\alpha&lt;\lambda\alpha+\mu\beta&lt;\beta$ or $\alpha>\lambda\alpha+\mu\beta>\beta$.<br> Given a Bruhat path $\Delta=(a_0,a_1,\cdots,a_r)$ and a reflection ordering $a_{i}^{-1}a_{i+1}}$, and $R_&lt;(u,v)=\sum_{\Delta\in B(u,v):D(\Delta,&lt;)=\emptyset}{q^{l(\Delta)}}$, where $u,v\in W$, $B(u,v)$ denotes all Bruhat paths from $u$ to $v$. Then Theorem 5.3.4 in Bjorner and Benti' book shows that $\tilde{R}{u,v}(q)=R_&lt;(u,v)$, where $q^{\frac{1}{2}l(u,v)}\tilde{R}{u,v}(q^{\frac{1}{2}}-q^{-\frac{1}{2}})=R_{u,v}(q)$. (I am sorry about the last two subscripts which I can't correctly write down!)</p> http://mathoverflow.net/questions/116714/are-there-any-binomial-poset-which-has-non-isomorphic-interval-of-the-same-length Are there any binomial poset which has non-isomorphic interval of the same length? Michael Zhong 2012-12-18T15:28:47Z 2012-12-20T02:44:34Z <p>Definition: A poset $P$ is called a binomial poset if it satisfy a. $P$ is locally finite with a $\hat{0}$, and contains a infinite chain. b. Every interval $[x, y]$ of $P$ is graded. If $l(x,y)$ = n, then we call $[x,y]$ an n-interval. c. For all $n \in \mathbb{N}$, any two $n$-intervals contain the same number of maximal chains. </p> http://mathoverflow.net/questions/116716/why-is-there-a-unique-increasing-maximal-path-in-any-bruhat-interval-under-any-re/117905#117905 Comment by Michael Zhong Michael Zhong 2013-01-08T11:37:25Z 2013-01-08T11:37:25Z Thanks again! This induction can be used for the reflection order corresponding to $w_0=t_1\cdots t_k s_1\cdots s_k$(GTM 231 Exercise 5.20) when $W$ is finite. But for an arbitrary reflection ordering, I am not sure whether this reasoning still works. http://mathoverflow.net/questions/116716/why-is-there-a-unique-increasing-maximal-path-in-any-bruhat-interval-under-any-re/117905#117905 Comment by Michael Zhong Michael Zhong 2013-01-03T03:19:42Z 2013-01-03T03:19:42Z Thank you very much! This is really helpful. http://mathoverflow.net/questions/116714/are-there-any-binomial-poset-which-has-non-isomorphic-interval-of-the-same-length/116836#116836 Comment by Michael Zhong Michael Zhong 2012-12-21T16:07:43Z 2012-12-21T16:07:43Z Thank you very much!