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In [[mathematics]], for a [[Lie group]] <math>G</math>, the [[Kirillov orbit method]] gives a heuristic method in [[representation theory]]. It connects the [[Fourier transform]]s of [[coadjoint orbit]]s, which lie in the [[dual space]] of the [[Lie algebra]] of ''G'', to the [[infinitesimal character]]s of the [[irreducible representation]]s. The method got its name after the [[Russia]]n mathematician [[Alexandre Kirillov]].
 
At its simplest, it states that a character of a Lie group may be given by the [[Fourier transform]] of the [[Dirac delta function]] [[support (mathematics)|support]]ed on the coadjoint orbits, weighted by the square-root of the [[Jacobian]] of the [[exponential map]], denoted by <math>j</math>. It does not apply to all Lie groups, but works for a number of classes of [[connected space|connected]] Lie groups, including [[nilpotent]], some [[Semisimple Lie group|semisimple]] groups, and [[compact group]]s.
 
The Kirillov orbit method has led to a number of important developments in Lie theory, including the [[Duflo isomorphism]] and the [[wrapping map]].
 
== Character formula for compact Lie groups ==
 
Let <math>\lambda</math> be the [[highest weight]] of an [[Group representation|irreducible representation]] in the [[dual]] of the [[Lie algebra]] of the [[maximal torus]], denoted by <math>\mathfrak{t}^*</math>, and <math>\rho</math> half the sum of the positive [[root of a function|roots]].
 
We denote by
 
:<math>\mathcal{O}_{\lambda + \rho}</math>
 
the coadjoint orbit through
 
:<math>\lambda + \rho \in \mathfrak{t}^*</math>
 
and
 
:<math>\mu_{\lambda + \rho}</math>
 
is the <math>G</math>-invariant [[Measure (mathematics)|measure]] on
 
:<math>\mathcal{O}_{\lambda + \rho}</math>
 
with total mass
 
:<math>\dim \pi = d_\lambda</math>,
 
known as the [[Liouville measure]]. If <math>\chi_\pi = \chi_\lambda</math> is the character of a [[group representation|representation]], then''' Kirillov's character formula''' for compact Lie groups is then given by
 
:<math> j(X) \chi_\lambda (\exp X) = \int_{\mathcal{O}_{\lambda + \rho}} e^{i\beta (X)}d\mu_{\lambda + \rho} (\beta), \; \forall \; X \in \mathfrak{g} </math>
 
== Example: SU(2) ==
 
For the case of [[SU(2)]], the [[highest weight]]s are the positive half integers, and <math> \rho = 1/2 </math>. The coadjoint orbits are the two-dimensional [[spheres]] of radius <math> \lambda + 1/2 </math>, centered at the origin in 3-dimensional space. 
 
By the theory of [[Bessel function]]s, it may be shown that
 
:<math> \int_{\mathcal{O}_{\lambda + 1/2}} e^{i\beta (X)}d\mu_{\lambda + 1/2} (\beta) = \frac{\sin((2\lambda + 1)X)}{X/2}, \; \forall \; X \in \mathfrak{g}, </math>
 
and
 
:<math> j(X) = \frac{\sin X/2}{X/2} </math>
 
thus yielding the characters of ''SU''(2):
 
:<math> \chi_\lambda (\exp X) = \frac{\sin((2\lambda + 1)X)}{\sin X/2} </math>
 
== References ==
*Kirillov, A. A., ''Lectures on the Orbit Method'', Graduate studies in Mathematics, 64, AMS, Rhode Island, 2004.
 
[[Category:Representation theory of Lie groups]]

Revision as of 08:18, 29 January 2014

In mathematics, for a Lie group G, the Kirillov orbit method gives a heuristic method in representation theory. It connects the Fourier transforms of coadjoint orbits, which lie in the dual space of the Lie algebra of G, to the infinitesimal characters of the irreducible representations. The method got its name after the Russian mathematician Alexandre Kirillov.

At its simplest, it states that a character of a Lie group may be given by the Fourier transform of the Dirac delta function supported on the coadjoint orbits, weighted by the square-root of the Jacobian of the exponential map, denoted by j. It does not apply to all Lie groups, but works for a number of classes of connected Lie groups, including nilpotent, some semisimple groups, and compact groups.

The Kirillov orbit method has led to a number of important developments in Lie theory, including the Duflo isomorphism and the wrapping map.

Character formula for compact Lie groups

Let λ be the highest weight of an irreducible representation in the dual of the Lie algebra of the maximal torus, denoted by t*, and ρ half the sum of the positive roots.

We denote by

𝒪λ+ρ

the coadjoint orbit through

λ+ρt*

and

μλ+ρ

is the G-invariant measure on

𝒪λ+ρ

with total mass

dimπ=dλ,

known as the Liouville measure. If χπ=χλ is the character of a representation, then Kirillov's character formula for compact Lie groups is then given by

j(X)χλ(expX)=𝒪λ+ρeiβ(X)dμλ+ρ(β),Xg

Example: SU(2)

For the case of SU(2), the highest weights are the positive half integers, and ρ=1/2. The coadjoint orbits are the two-dimensional spheres of radius λ+1/2, centered at the origin in 3-dimensional space.

By the theory of Bessel functions, it may be shown that

𝒪λ+1/2eiβ(X)dμλ+1/2(β)=sin((2λ+1)X)X/2,Xg,

and

j(X)=sinX/2X/2

thus yielding the characters of SU(2):

χλ(expX)=sin((2λ+1)X)sinX/2

References

  • Kirillov, A. A., Lectures on the Orbit Method, Graduate studies in Mathematics, 64, AMS, Rhode Island, 2004.