<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hi everyone,\n\nCould anybody help me in showing the equivalence of the two\ndefinitions of the intetwiners (used in spin networks)? What I mean is\nthat there are two ways in the literature to define them:\nAs invariant tensors in the tensor product of the representations\nlabelling the edges.\nAs a linear map from the tensor product of the reps on the incoming\nedges to the reps on the outgoing edges.\n\nHow are they related? It seems that there is no invariance postulated\nin the second definition. And there is no mentioning of the\nincoming-outgoing in the first.\n\nThank you very much for your help.\nT.T.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi everyone,
Could anybody help me in showing the equivalence of the two
definitions of the intetwiners (used in spin networks)? What I mean is
that there are two ways in the literature to define them:
As invariant tensors in the tensor product of the representations
labelling the edges.
As a linear map from the tensor product of the reps on the incoming
edges to the reps on the outgoing edges.
How are they related? It seems that there is no invariance postulated
in the second definition. And there is no mentioning of the
incoming-outgoing in the first.
Thank you very much for your help.
T.T.
Greg Egan
Apr24-04, 11:15 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <856a4a19.0404201513.55cf46fc@posting.google.com>, \ntimtlas@hotmail.com (Tlas) wrote:\n\n> Could anybody help me in showing the equivalence of the two\n> definitions of the intetwiners (used in spin networks)? What I mean is\n> that there are two ways in the literature to define them:\n> As invariant tensors in the tensor product of the representations\n> labelling the edges.\n> As a linear map from the tensor product of the reps on the incoming\n> edges to the reps on the outgoing edges.\n>\n> How are they related? It seems that there is no invariance postulated\n> in the second definition. And there is no mentioning of the\n> incoming-outgoing in the first.\n\nIn the second definition, you really need to say "a linear map that\ncommutes with the action of the group representation".\n\nFor example, suppose the group is G, and V_1 ... V_4 are vector spaces\nequipped with representations rho_1 ... rho_4 of G. Then an intertwiner\nfrom V_1 tensor V_2 to V_3 tensor V_4 is a fixed linear map (i.e.\nindependent of g, unlike the linear maps rho_i(g)) that satisfies the\nfollowing equation:\n\nT((rho_1 tensor rho_2)(g)(x))\n\n= (rho_3 tensor rho_4)(g)(T(x))\n\nfor all g in G, and all x in V_1 tensor V_2.\n\nHere, the representations on the tensor products are defined in the\nobvious way:\n\n(rho_1 tensor rho_2)(g)(x_1 tensor x_2)\n\n= rho_1(g)(x_1) tensor rho_2(g)(x_2)\n\nfor all x_1 in V_1 and x_2 in V_2, and then linearity extends this\ndefinition to all elements of V_1 tensor V_2.\n\nNow, any linear operator from a vector space V to a vector space W can be\nthought of as a tensor in the space V* tensor W, where V* is the dual\nspace to V (i.e. the space of scalar-valued linear functions on V). If\n\nf belongs to V*\nw belongs to W\n\nthen we can define a linear map from V to W by:\n\n(f tensor w)(v) = f(v) w\n\nand we can also write any linear map T from V to W as a sum of these\nkinds of maps. Say {e_i} is a basis of V, {f^i} is the dual basis of V*\n(with f^j(e_i)=delta^j_i), {d_i} is a basis of W, and {h^i} is the dual\nbasis of W* (with h^j(d_i)=delta^j_i).\n\nDefine T^j_i = h^j(T(e_i)), so T(e_i) = T^j_i d_j\n\nIf v = v^i e_i we have v^i = f^i(v)\n\nT(v) = T(f^i(v) e_i)\n= f^i(v) T(e_i)\n= f^i(v) T^j_i d_j\n= T^j_i f^i(v) d_j\n\nIf you write out the relationship for T commuting with the group action,\nyou can derive an expression for how the coordinates of T transform.\n\nNote that T is a tensor in V* tensor W, not V tensor W. For some\nrepresentations (such as the spin representations of SU(2)) the dual\nrepresentation is equivalent to the original representation, and you can\nuse the linear map that proves the equivalence to identify the two\nspaces. But in general that won\'t be the case.\n\nThe dual rep, (rho*, on the space V*), of a rep rho on V is defined by:\n\n(rho*(g)(f))(v) = f(rho(g^{-1})v)\n\nfor f in V*, v in V.\n\nFor the spin reps, the dual reps are equivalent, i.e. there is a\nbijective intertwiner from V to V*. In the case of the spin-(1/2) rep,\nthis intertwiner has the coordinates:\n\nT_{ij} = | 0 1 |\n| -1 0 |\n\nThe relationship we need to prove that the spin-(1/2) rep and its dual\nare equivalent is:\n\nrho*(g) T = T rho(g)\n\nfor all g in SU(2), i.e. in coordinate form:\n\n[rho*(g)]^b_a T_{bc} = T_{ab} [rho(g)]^b_c\n[g^{-1}]^b_a T_{bc} = T_{ab} g^b_c\n\nMultiplying both sides by g^a_d:\n\ng^a_d [g^{-1}]^b_a T_{bc} = T_{ab} g^a_d g^b_c\nT_{dc} = T_{ab} g^a_d g^b_c\n\nYou can check that this is the case for any element of SU(2), as the RHS\njust gives you either 0, or +/- the determinant of g, which is 1.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <856a4a19.0404201513.55cf46fc@posting.google.com>,
timtlas@hotmail.com (Tlas) wrote:
> Could anybody help me in showing the equivalence of the two
> definitions of the intetwiners (used in spin networks)? What I mean is
> that there are two ways in the literature to define them:
> As invariant tensors in the tensor product of the representations
> labelling the edges.
> As a linear map from the tensor product of the reps on the incoming
> edges to the reps on the outgoing edges.
>
> How are they related? It seems that there is no invariance postulated
> in the second definition. And there is no mentioning of the
> incoming-outgoing in the first.
In the second definition, you really need to say "a linear map that
commutes with the action of the group representation".
For example, suppose the group is G, and V_1 ... V_4 are vector spaces
equipped with representations \rho_1 ... \rho_4 of G. Then an intertwiner
from V_1 tensor V_2 to V_3 tensor V_4 is a fixed linear map (i.e.
independent of g, unlike the linear maps \rho_i(g)) that satisfies the
following equation:
T((\rho_1[/itex] tensor \rho_2)(g)(x))= (\rho_3 tensor \rho_4)(g)(T(x))
for all g in G, and all x in V_1 tensor V_2.
Here, the representations on the tensor products are defined in the
obvious way:
(\rho_1 tensor \rho_2)(g)(x_1 tensor x_2)= \rho_1(g)(x_1) tensor \rho_2(g)(x_2)
for all x_1 in V_1 and x_2 in V_2, and then linearity extends this
definition to all elements of V_1 tensor V_2.
Now, any linear operator from a vector space V to a vector space W can be
thought of as a tensor in the space V* tensor W, where V* is the dual
space to V (i.e. the space of scalar-valued linear functions on V). If
f belongs to V*
w belongs to W
then we can define a linear map from V to W by:
(f tensor w)(v) = f(v) w
and we can also write any linear map T from V to W as a sum of these
kinds of maps. Say {e_i} is a basis of V, {f^i} is the dual basis of V*
(with f^j(e_i)=\delta^j_i), {d_i} is a basis of W, and {h^i} is the dual
basis of W* (with h^j(d_i)=\delta^j_i).
Define T^{j_i} = h^j(T(e_i)), so T(e_i) = T^{j_i} d_j
If v = v^i e_i we have v^i = f^i(v)
T(v) = T(f^i(v) e_i)= f^i(v) T(e_i)= f^i(v) T^{j_i} d_j= T^{j_i} f^i(v) d_j
If you write out the relationship for T commuting with the group action,
you can derive an expression for how the coordinates of T transform.
Note that T is a tensor in V* tensor W, not V tensor W. For some
representations (such as the spin representations of SU(2)) the dual
representation is equivalent to the original representation, and you can
use the linear map that proves the equivalence to identify the two
spaces. But in general that won't be the case.
The dual rep, (\rho*, on the space V*), of a rep \rho on V is defined by:
(\rho*(g)(f))(v) = f(\rho(g^{-1})v)
for f in V*, v in V.
For the spin reps, the dual reps are equivalent, i.e. there is a
bijective intertwiner from V to V*. In the case of the spin-(1/2) rep,
this intertwiner has the coordinates:
T_{ij} = | 1 |
| -1 |
The relationship we need to prove that the spin-(1/2) rep and its dual
are equivalent is:
[itex]\rho*(g) T = T \rho(g)
for all g in SU(2), i.e. in coordinate form:
[\rho*(g)]^b_a T_{bc} = T_{ab} [\rho(g)]^b_c[g^{-1}]^b_a T_{bc} = T_{ab} g^{b_c}
Multiplying both sides by g^{a_d}:g^{a_d} [g^{-1}]^b_a T_{bc} = T_{ab} g^{a_d} g^{b_c}T_{dc} = T_{ab} g^{a_d} g^{b_c}
You can check that this is the case for any element of SU(2), as the RHS
just gives you either 0, or +/- the determinant of g, which is 1.
John Baez
May26-04, 05:12 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <856a4a19.0404201513.55cf46fc@posting.google.com>, \nTlas <timtlas@hotmail.com> wrote:\n\n>Could anybody help me in showing the equivalence of the two\n>definitions of the intetwiners (used in spin networks)? What I mean is\n>that there are two ways in the literature to define them:\n>As invariant tensors in the tensor product of the representations\n>labelling the edges.\n>As a linear map from the tensor product of the reps on the incoming\n>edges to the reps on the outgoing edges.\n\n>How are they related? It seems that there is no invariance postulated\n>in the second definition.\n\nThat\'s because you\'re not stating the second definition correctly.\nThe correct definition is that it\'s an linear map from the tensor\nproduct of the reps on the incoming edges to the reps on the outgoing\nedges, say\n\nT: V_1 tensor ... tensor V_n -> V\'_1 tensor V\'_m\n\nSUCH THAT:\n\nT ((rho_1(g) tensor ... tensor rho_n(g)) =\n(rho\'_1(g) tensor ... tensor rho\'_n(g)) T\n\nHere we have representations rho_i on the vector spaces V_i,\nand rho\'_i on the vector spaces V\'_i.\n\nThis condition is invariance in disguise.\n\n>And there is no mentioning of the\n>incoming-outgoing in the first.\n\nThat\'s because you can use duality to take any linear operator like\nthis:\n\nT: V_1 tensor ... tensor V_n -> V\'_1 tensor V\'_m\n\nand turn it into one like this:\n\nS: C -> (V_1)* tensor ... tensor (V_n)* tensor V\'_1 tensor ... tensor V\'_m\n\nbut a linear map like this contains the same information as the vector\n\nS(1) in (V_1)* tensor ... tensor (V_n)* tensor V\'_1 tensor ... tensor V\'_m\n\nand T is an intertwiner iff S(1) is invariant... so the two\ndefinitions are equivalent.\n\nIt sounds like you\'re diving into spin networks without any\nprevious experience in group representation theory - e.g.\nfamiliarity with the definition of "intertwining operator"\nor the use of duality. I can give you some references if you\nwant to bone up on that sutff.\n\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <856a4a19.0404201513.55cf46fc@posting.google.com>,
Tlas <timtlas@hotmail.com> wrote:
>Could anybody help me in showing the equivalence of the two
>definitions of the intetwiners (used in spin networks)? What I mean is
>that there are two ways in the literature to define them:
>As invariant tensors in the tensor product of the representations
>labelling the edges.
>As a linear map from the tensor product of the reps on the incoming
>edges to the reps on the outgoing edges.
>How are they related? It seems that there is no invariance postulated
>in the second definition.
That's because you're not stating the second definition correctly.
The correct definition is that it's an linear map from the tensor
product of the reps on the incoming edges to the reps on the outgoing
edges, say
T: V_1[/itex] tensor ... tensor V_n -> V'_1 tensor V'_m
SUCH THAT:
T ((\rho_1(g) tensor ... tensor \rho_n(g)) =(\rho'_1(g) tensor ... tensor \rho'_n(g)) T
Here we have representations \rho_i on the vector spaces V_i,
and \rho'_i on the vector spaces V'_i.
This condition is invariance in disguise.
>And there is no mentioning of the
>incoming-outgoing in the first.
That's because you can use duality to take any linear operator like
this:
T: V_1 tensor ... tensor V_n -> V'_1 tensor [itex]V'_m
and turn it into one like this:
S: C -> (V_1)* tensor ... tensor (V_n)* tensor V'_1 tensor ... tensor V'_m
but a linear map like this contains the same information as the vector
S(1) in (V_1)* tensor ... tensor (V_n)* tensor V'_1 tensor ... tensor V'_m
and T is an intertwiner iff S(1) is invariant... so the two
definitions are equivalent.
It sounds like you're diving into spin networks without any
previous experience in group representation theory - e.g.
familiarity with the definition of "intertwining operator"
or the use of duality. I can give you some references if you
want to bone up on that sutff.
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