Split Lie algebra: Difference between revisions

Revision as of 04:04, 12 February 2013

In geometry, Hessenberg varieties, first studied by De Mari, Procesi, and Shayman, are a family of subvarieties of the full flag variety which are defined by a Hessenberg function h and a linear transformation X. The study of Hessenberg varieties was first motivated by questions in numerical analysis in relation to algorithms for computing eigenvalues and eigenspaces of the linear operator X. Later work by Springer, Peterson, Kostant, among others, found connections with combinatorics, representation theory and cohomology.

Definitions

A Hessenberg function is a function of tuples

h : {1, 2, \dots, n} \to {1, 2, \dots, n}

where

h (i + 1) \geq max (i, h (i)) for all 1 \leq i \leq n - 1 .

For example,

h (1, 2, 3, 4, 5) = (2, 3, 3, 4, 5)

is a Hessenberg function.

For any Hessenberg function h and a linear transformation

X : ℂ^{n} \to ℂ^{n},

the Hessenberg variety is the set of all flags $F_{∙}$ such that

X \cdot F_{i} \subseteq F_{(h (i))}

for all i. Here $F_{(h (i))}$ denotes the vector space spanned by the first $h (i)$ vectors in the flag $F_{∙}$ .

ℋ (X, h) = {F_{∙} ∣ X F_{i} \subset F_{(h_{i})} for 1 \leq i \leq n}

Examples

Some examples of Hessenberg varieties (with their $h$ function) include:

The Full Flag variety: h(i) = n for all i

The Peterson variety: $h (i) = i + 1$ for $i = 1, 2, \dots, n - 1$

The Springer variety: $h (i) = i$ for all $i$ .

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

F. De Mari, C. Procesi, and M. Shayman, Hessenberg varieties, Trans. Amer. Math. Soc. 332 (1992), 529–534.
B. Kostant, Flag Manifold Quantum Cohomology , the Toda Lattice, and the Representation with Highest Weight $ρ$ , Selecta Mathematica. (N.S.) 2, 1996, 43–91.
J. Tymoczko, Linear conditions imposed on flag varieties, Amer. J. Math. 128 (2006), 1587–1604.

@@ Line 1: / Line 1: @@
-Surely the second option would be more beneficial for any website. It is very easy to customize plugins according to the needs of a particular business. The Word - Press Dashboard : an administrative management tool that supports FTP content upload  2. Word - Press also provides protection against spamming, as security is a measure issue.  If you have any type of concerns concerning where and ways to make use of [http://247url.info/wordpress_backup_239161 wordpress backup], you can call us at the web site. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>
+In [[geometry]], '''Hessenberg varieties''', first studied by De Mari, [[Claudio Procesi|Procesi]], and Shayman, are a family of [[subvarieties]] of the full [[flag variety]] which are defined by a Hessenberg function ''h'' and a linear transformation&nbsp;''X''.  The study of Hessenberg varieties was first motivated by questions in [[numerical analysis]] in relation to algorithms for computing eigenvalues and eigenspaces of the linear operator&nbsp;''X''.  Later work by Springer, Peterson, Kostant, among others, found connections with [[combinatorics]], [[representation theory]] and [[cohomology]].
-Always remember that an effective linkwheel strategy strives to answer all the demands of popular  search engines while reacting to the latest marketing number trends. Some of the Wordpress development services offered by us are:. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Word - Press makes it possible to successfully and manage your website. <br><br>Usually, Wordpress owners selling the ad space on monthly basis and this means a residual income source. As of now, Pin2Press is getting ready to hit the market. You've got invested a great cope of time developing and producing up the topic substance. These frequent updates have created menace in the task of optimization. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait. <br><br>The primary differences are in the plugins that I install, as all sites don't need all the normal plugins. High Quality Services: These companies help you in creating high quality Word - Press websites. Enterprise, when they plan to hire Word - Press developer resources still PHP, My - SQL and watch with great expertise in codebase. Contact Infertility Clinic Providing One stop Fertility Solutions at:. OSDI, a  Wordpress Development Company  based on ahmedabad, India. <br><br>As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Visit our website to learn more about how you can benefit. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. As with a terminology, there are many methods to understand how to use the terminology.
+== Definitions ==
+A ''Hessenberg function'' is a function of tuples
+:<math>h :\{1,2, \ldots,n \} \rightarrow \{1,2, \ldots,n \}</math>
+where
+:<math> h(i+1) \geq \text{max }(i,h(i))  \text{ for all} 1 \leq i \leq n-1. </math>
+For example,
+:<math> h(1,2,3,4,5)=(2,3,3,4,5) \,  </math>
+is a Hessenberg function.
+For any Hessenberg function ''h'' and a linear transformation
+:<math> X: \C^n \rightarrow \C^n, \, </math>
+the ''Hessenberg variety'' is the set of all flags <math> F_{\bullet} </math> such that
+:<math> X \cdot F_i \subseteq F_{(h(i))} </math>
+for all i.  Here <math> F_{(h(i))} </math> denotes the vector space spanned by the first <math> h(i) </math> vectors in the flag <math> F_{\bullet} </math>.
+:<math> \mathcal{H}(X,h)  = \{ F_{\bullet} \mid X F_{i} \subset F_{(h_i)} \text{ for } 1 \leq i \leq n \} </math>
+== Examples ==
+Some examples of Hessenberg varieties (with their <math>h</math> function) include:
+The Full Flag variety: ''h''(''i'') = ''n'' for all ''i''
+The [[Peterson variety]]: <math>h(i) = i+1</math> for <math> i = 1,2,\dots, n-1</math>
+The [[Springer variety]]: <math> h(i) = i </math> for all <math> i </math>.
+==References==
+{{Reflist}}
+*F. De Mari, [[Claudio Procesi|C. Procesi]], and M. Shayman, ''Hessenberg varieties'', Trans. Amer. Math. Soc. 332 (1992), 529–534.
+*[[B. Kostant]], ''Flag Manifold Quantum Cohomology , the Toda Lattice, and the Representation with Highest Weight <math> \rho </math>,'' Selecta Mathematica. (N.S.) '''2''', 1996, 43&ndash;91.
+*J. Tymoczko, ''Linear conditions imposed on flag varieties'', Amer. J. Math. 128 (2006), 1587–1604.
+[[Category:Algebraic geometry]]
+[[Category:Algebraic combinatorics]]

Split Lie algebra: Difference between revisions

Revision as of 04:04, 12 February 2013

Definitions

Examples

References

Navigation menu

Search