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Dec
2
awarded  Notable Question
Nov
6
comment The longest word in Weyl group and positive roots.
I was asking a similar question in here math.stackexchange.com/questions/1513075/…. If you have time plz take a look.
Nov
6
comment The longest word in Weyl group and positive roots.
Yes, you're correct by saying that I'm mentioning "height" of positive roots. My recently question is exact what you say if given reduced expression $w=s_{1}s_{2}\cdots s_{r}$ ($s_{i}$ simple reflection w.r.t simple $\alpha_i$) then $\beta_i:=s_rs_{r-1}\ldots s_{i+1}(\alpha_i), \beta_r:=\alpha_r$ are positive roots sent negative by $w$. I wanted to know that can we order them as $\beta_r<\beta_{r-1}<\ldots<\beta_1$ with $<$ I mean the order I defined previously?
Nov
6
comment The longest word in Weyl group and positive roots.
@HughDenoncourt Thank you. In your first comment, by $"<"$ you meant "reflection ordering" that defined in Bjorner and Brenti's textbook? Isn't it the usual lexicographic order that $\alpha \le \beta \iff \beta -\alpha$ is a linear combination of simple roots with non-negative coefficients?
Nov
5
comment The longest word in Weyl group and positive roots.
@HughDenoncourt Can you give me the references of ordering roots in your first comment?
Nov
4
asked A chain between two positive roots having same length
Nov
3
asked Definitions of length function on a Weyl group
Nov
2
comment Roots of height 1 are necessarily simple.
Is it simply a solving equation $ \sum\limits_{i=1}^{l} \lambda_i=1$ over non-negative integers?
Nov
2
answered Roots of height 1 are necessarily simple.
Nov
1
comment Lie algebras and roots systems
@JyrkiLahtonen Thanks for the neat proof. In your ordered sequence $\{\alpha_k\}$, is it true that $\alpha_k-\alpha_{k+1}$ is a simple root for every $k=0,1,\ldots, \ell -1$?
Oct
17
accepted $f(x)$ has factor $x-a$ iff $R$ is UFD?
Oct
17
asked $A \oplus M$ is Noetherian iff $A$ is Noetherian and $M$ is a f.g. $A$-module
Oct
16
accepted $\phi(X)$ is essential and $\psi \circ \phi$ is injective then $\psi$ is injective.
Oct
16
comment $\phi(X)$ is essential and $\psi \circ \phi$ is injective then $\psi$ is injective.
@PrahladVaidyanathan wow! I like your one-line proof! Is there any way to find a proof with my posted approach?
Oct
16
revised $\phi(X)$ is essential and $\psi \circ \phi$ is injective then $\psi$ is injective.
edited title
Oct
16
asked $\phi(X)$ is essential and $\psi \circ \phi$ is injective then $\psi$ is injective.
Oct
15
asked $f(x)$ has factor $x-a$ iff $R$ is UFD?
Oct
15
accepted Automorphism of order 2 of C[x,y] and its ring of invariants
Oct
15
comment Automorphism of order 2 of C[x,y] and its ring of invariants
Do you mean that $x^my^n=(xy)^n(x^{m-n}-y^{m-n})+x^ny^m$? I still don't get the answer..
Oct
15
comment Automorphism of order 2 of C[x,y] and its ring of invariants
How can you prove the mentioned fact above? In particular how to write $x^my^n \in A$ as $1\cdot f_1 +x\cdot f_2$?