Singular point of an algebraic variety: Difference between revisions

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In [[mathematics]], an (''H'',''K'') '''double coset''' in ''G'', where ''G'' is a [[group (mathematics)|group]] and ''H'' and ''K'' are [[subgroup]]s of ''G'', is an [[equivalence class]] for the [[equivalence relation]] defined on ''G'' by


:''x'' ~ ''y'' if there are ''h'' in ''H'' and ''k'' in ''K'' with ''hxk'' = ''y''.
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Then each double coset is of the form ''HxK'', and ''G'' is partitioned into its (''H'',''K'') double cosets; each of them is a union of ordinary right [[coset]]s ''Hy'' of ''H'' in ''G'' and left cosets ''zK'' of K in G. In another aspect, these are in fact [[orbit (group theory)|orbit]]s for the group action of ''H''×''K'' on ''G'' with ''H'' acting by left multiplication and ''K'' by inverse right multiplication. The set of double cosets can be written
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:''H''\''G''/''K''.
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== Algebraic structure ==
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It is possible to define a product operation of double cosets in an associated ring.


Given two double cosets <math> H g_1 K</math> and <math>K g_2 L</math>, we decompose each into right cosets <math> H g_1 K = \coprod_i H a_i</math> and <math> K g_2 L = \coprod_j b_j L</math>. If we write <math> c_g = \left | \{ (i,j) : a_i b_j \in H g \} \right |</math>, then we can define the product of <math> H g_1 K</math> with <math> K g_2 L</math> as the formal sum
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:<math> H g_1 K \cdot K g_2 L = \sum_{g \in H \backslash G} c_g H g L.</math>
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An important case is when ''H'' = ''K'' = ''L'', which allows us to define an algebra structure on the associated ring spanned by linear combinations of double cosets.
 
== Applications ==
Double cosets are important in connection with [[representation theory]], when a representation of ''H'' is used to construct an [[induced representation]] of ''G'', which is then [[restricted representation|restricted]] to ''K''. The corresponding double coset structure carries information about how the resulting representation decomposes.
 
They are also important in [[functional analysis]], where in some important cases functions left-invariant and right-invariant by a subgroup ''K'' can form a [[commutative ring]] under [[convolution]]: see [[Gelfand pair]].
 
In geometry, a [[Clifford–Klein form]] is a double coset space ''Γ''\''G''/''H'', where ''G'' is a [[reductive Lie group]], ''H'' is a closed subgroup, and ''Γ'' is a discrete subgroup (of ''G'') that acts [[properly discontinuously]] on the [[homogeneous space]] ''G''/''H''.
 
In [[number theory]], the [[Hecke operator|Hecke algebra]] corresponding to a [[congruence subgroup]] ''Γ'' of the [[modular group]] is spanned by elements of the double coset space <math>\Gamma \backslash \mathrm{GL}_2^+(\mathbb{Q}) / \Gamma</math>; the algebra structure is that acquired from the multiplication of double cosets described above. Of particular importance are the Hecke operators <math>T_m</math> corresponding to the double cosets <math>\Gamma_0(N) g \Gamma_0(N)</math> or <math>\Gamma_1(N) g \Gamma_1(N)</math>, where <math>g= \begin{pmatrix} 1 & 0 \\ 0 & m \end{pmatrix}</math> (these have different properties depending on whether ''m'' and ''N'' are coprime or not), and the diamond operators <math> \langle d \rangle</math> given by the double cosets <math>\Gamma_1(N) \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Gamma_1(N)</math> where <math> d \in (\mathbb{Z}/N\mathbb{Z})^\times</math> and we require <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)</math> (the choice of ''a'', ''b'', ''c'' does not affect the answer).  
 
 
{{DEFAULTSORT:Double Coset}}
[[Category:Group theory]]

Latest revision as of 23:25, 9 June 2014

The Saboteur is able to force opponents to trash an expensive card and replace it with something cheaper. Here are few best free Naruto games online that will mostly appeal kids and teens:. That assumes you're really working and not just coasting along. A new card type - Duration cards - is also introduced in this set, changing the way games are played. What is it about online variations of Zelda games that make them so beneficial to your brain. What IS crucial is the slope the club is placed in your hand. 

It’s more on doing something that will lead you to a certain end goal. The reason why is because instead of just running around blowing things up, you actually have to solve a puzzle while at the same time enjoying an action game. You can also hire new employees, bribe other players for their abilities, and use cash to buy up shares, departments and external consultants. After moving the coins associated with the two dice, if there are two or more coins lined up in a row, other than 1 and 7, then the player may take them off the board.

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