15-j sybmols

T.M.Tlas@gmail.com

<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\nI was hoping that some expert ;) may enlighten me on this issue.\n\nI\'ve been reading a lot about the Crane-Yetter TQFT lately, and it\nseems all constructions of it (or of isomorphic TQFT\'s) use some\nordering of vertices at an intermediate step. The question is Why?\n\nIs this because the graph corresponding to a 4-simplex is not\nembeddable in 2d without self intersections? So that if we just embedd\nit randomly in some way, the diagramme (number) corresponding to it\nwill be different from the one obtained by embedding it some other way.\n\nIf the answer is yes to above, why that particular convention is\nchosen? Are there any other consistent conventions for embedding the\ndiagramme? (Just to show that i\'m totally spoilt, why is the 4 simplex,\nfor example, put on the "right" of the 0 and the 2 simplex on the left?\nI\'m talking about the diagramme in the paper by Crane and Yetter "A\ncategorical construction of 4d topological quantum field theories")\n\nFinally, I\'ve seen MANY books, papers, articles ,etc., discussing the\n6-j symbols, their relationship to the tetrahedron, identities among\nthem etc., but have never seen ANY book which discusses the 15-j ones.\nCan anybody point to a reference which does, and ARE there similar\nidentities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..)\n\nThank you for your time,\nTim\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,

I was hoping that some expert ;) may enlighten me on this issue.

I've been reading a lot about the Crane-Yetter TQFT lately, and it
seems all constructions of it (or of isomorphic TQFT's) use some
ordering of vertices at an intermediate step. The question is Why?

Is this because the graph corresponding to a 4-simplex is not
embeddable in 2d without self intersections? So that if we just embedd
it randomly in some way, the diagramme (number) corresponding to it
will be different from the one obtained by embedding it some other way.

If the answer is yes to above, why that particular convention is
chosen? Are there any other consistent conventions for embedding the
diagramme? (Just to show that i'm totally spoilt, why is the 4 simplex,
for example, put on the "right" of the and the 2 simplex on the left?
I'm talking about the diagramme in the paper by Crane and Yetter "A
categorical construction of 4d topological quantum field theories")

Finally, I've seen MANY books, papers, articles ,etc., discussing the
6-j symbols, their relationship to the tetrahedron, identities among
them etc., but have never seen ANY book which discusses the [itex]15-j[/itex] ones.
Can anybody point to a reference which does, and ARE there similar
identities in the [itex]15-j[/itex] case (Biedenharn-Elliot, orthogonality, etc..)

Thank you for your time,
Tim

Hsuan-Yeh Chang

<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><snip>\n\n> Finally, I\'ve seen MANY books, papers, articles ,etc., discussing the\n> 6-j symbols, their relationship to the tetrahedron, identities among\n> them etc., but have never seen ANY book which discusses the 15-j ones.\n> Can anybody point to a reference which does, and ARE there similar\n> identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..)\n>\n> Thank you for your time,\n> Tim\n\n3j -> coupling of two angular momentum\n6j -> coupling of three angular momentum\n9j -> coupling of four angular momentum\n12j -> coupling of five angular momentum\n15j -> coupling of six angular momentum\n\nI believe you can derive your 15j symbols, if you really need one, by\nfollowing the rules of the addition of angular momentum, only a lot more\ncomplicated...\n\nGood Luck!\n\nHYC\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><snip>

> Finally, I've seen MANY books, papers, articles ,etc., discussing the
> 6-j symbols, their relationship to the tetrahedron, identities among
> them etc., but have never seen ANY book which discusses the [itex]15-j[/itex] ones.
> Can anybody point to a reference which does, and ARE there similar
> identities in the [itex]15-j[/itex] case (Biedenharn-Elliot, orthogonality, etc..)
>
> Thank you for your time,
> Tim

[itex]3j ->[/itex] coupling of two angular momentum
[itex]6j ->[/itex] coupling of three angular momentum
[itex]9j ->[/itex] coupling of four angular momentum
12j -> coupling of five angular momentum
15j -> coupling of six angular momentum

I believe you can derive your 15j symbols, if you really need one, by
following the rules of the addition of angular momentum, only a lot more
complicated...

Good Luck!

HYC

Igor Khavkine

<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>T.M.Tlas@gmail.com wrote:\n> Hi Everyone,\n>\n> I was hoping that some expert ;) may enlighten me on this issue.\n\nI don\'t know if I qualify as an expert, but I did spend a bit of time\nlooking at the paper by Crane, Kauffman and Yetter (hep-th/9409167).\n\n> I\'ve been reading a lot about the Crane-Yetter TQFT lately, and it\n> seems all constructions of it (or of isomorphic TQFT\'s) use some\n> ordering of vertices at an intermediate step. The question is Why?\n>\n> Is this because the graph corresponding to a 4-simplex is not\n> embeddable in 2d without self intersections? So that if we just\n> embedd it randomly in some way, the diagramme (number) corresponding\n> to it will be different from the one obtained by embedding it some\n> other way.\n\nBasically yes. an ordering on the vertices allows one to canonically\nconstruct the 15j symbol with the appropriate intersections. The\nintersections are important in the q-deformed case. Specifically, if\nyou look at the diagram on page 21 of the paper I referenced above,\nit\'s construction (unfortunately rather opaquely) is described starting\nwith the last three paragraphs on page 22.\n\n> If the answer is yes to above, why that particular convention is\n> chosen? Are there any other consistent conventions for embedding the\n> diagramme? (Just to show that i\'m totally spoilt, why is the 4\n> simplex, for example, put on the "right" of the 0 and the 2 simplex\n> on the left? I\'m talking about the diagramme in the paper by Crane\n> and Yetter "A categorical construction of 4d topological quantum\n> field theories")\n\nUnfortunately, the version of the paper you mention that is on the\narXive (hep-th/9301062) doesn\'t have the figures. So I can\'t comment on\nthem. But I would guess that if you look at the figure on page 21 of\nthe paper I already mentioned, it\'s probably the same or a similar one.\nBesides the sort-of natural construction of this diagram as a\nprojection of a 4-symplex onto a 2D plane, its main merit is the fact\nthat it allows the authors to prove the invariance of their state sum\nunder change of triangulation. This is proved in a sequence of\ndiagramatic lemmas in the subsequent pages.\n\n> Finally, I\'ve seen MANY books, papers, articles ,etc., discussing the\n> 6-j symbols, their relationship to the tetrahedron, identities among\n> them etc., but have never seen ANY book which discusses the 15-j\n> ones. Can anybody point to a reference which does, and ARE there\n> similar identities in the 15-j case (Biedenharn-Elliot,\n> orthogonality, etc..)\n\nThe reason the 6j symbol is often discussed is because of its relation\nto recoupling theory. Diagramatically, the 6j symbol gives the\ncoefficients that allow us to rewrite\n\n\\ / \\ /\n\\____/ as linear combinations of \\ / .\n/ \\ |\n/ \\ |\n/ \\\n/ \\\n\nIn other words, you can view it as a sort of change of basis for the\nspace of intertwiners between pairs of representations of SU(2). Most\nof the identities and the 6j\'s evaluation in terms of the tetrahedral\nnetwork comes from this recoupling formula. As far as I know, the 15j\nsymbol is not associated with any such recoupling formulas. Thus, if\nthere are identities or recurrence relations for the 15j, they are not\nas easy to discover. Mostly, it\'s just a network with 15 edges that\ngives the right sort of amplitude in the Crane-Yetter topological state\nsum. That\'s its most important property.\n\nHope this helps.\n\nIgor\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>T.M.Tlas@gmail.com wrote:
> Hi Everyone,
>
> I was hoping that some expert ;) may enlighten me on this issue.

I don't know if I qualify as an expert, but I did spend a bit of time
looking at the paper by Crane, Kauffman and Yetter (http://www.arxiv.org/abs/hep-th/9409167).

> I've been reading a lot about the Crane-Yetter TQFT lately, and it
> seems all constructions of it (or of isomorphic TQFT's) use some
> ordering of vertices at an intermediate step. The question is Why?
>
> Is this because the graph corresponding to a 4-simplex is not
> embeddable in 2d without self intersections? So that if we just
> embedd it randomly in some way, the diagramme (number) corresponding
> to it will be different from the one obtained by embedding it some
> other way.

Basically yes. an ordering on the vertices allows one to canonically
construct the 15j symbol with the appropriate intersections. The
intersections are important in the q-deformed case. Specifically, if
you look at the diagram on page 21 of the paper I referenced above,
it's construction (unfortunately rather opaquely) is described starting
with the last three paragraphs on page 22.

> If the answer is yes to above, why that particular convention is
> chosen? Are there any other consistent conventions for embedding the
> diagramme? (Just to show that i'm totally spoilt, why is the 4
> simplex, for example, put on the "right" of the and the 2 simplex
> on the left? I'm talking about the diagramme in the paper by Crane
> and Yetter "A categorical construction of 4d topological quantum
> field theories")

Unfortunately, the version of the paper you mention that is on the
arXive (http://www.arxiv.org/abs/hep-th/9301062) doesn't have the figures. So I can't comment on
them. But I would guess that if you look at the figure on page 21 of
the paper I already mentioned, it's probably the same or a similar one.
Besides the sort-of natural construction of this diagram as a
projection of a 4-symplex onto a 2D plane, its main merit is the fact
that it allows the authors to prove the invariance of their state sum
under change of triangulation. This is proved in a sequence of
diagramatic lemmas in the subsequent pages.

> Finally, I've seen MANY books, papers, articles ,etc., discussing the
> 6-j symbols, their relationship to the tetrahedron, identities among
> them etc., but have never seen ANY book which discusses the [itex]15-j[/itex]
> ones. Can anybody point to a reference which does, and ARE there
> similar identities in the [itex]15-j[/itex] case (Biedenharn-Elliot,
> orthogonality, etc..)

The reason the 6j symbol is often discussed is because of its relation
to recoupling theory. Diagramatically, the 6j symbol gives the
coefficients that allow us to rewrite

[tex]\ / \ /[/itex]
\__{__}/ as linear combinations of [itex]\ / ./ \ |/ \ |[/itex]
/ \
[itex]/ \[/tex]

In other words, you can view it as a sort of change of basis for the
space of intertwiners between pairs of representations of SU(2). Most
of the identities and the 6j's evaluation in terms of the tetrahedral
network comes from this recoupling formula. As far as I know, the 15j
symbol is not associated with any such recoupling formulas. Thus, if
there are identities or recurrence relations for the 15j, they are not
as easy to discover. Mostly, it's just a network with 15 edges that
gives the right sort of amplitude in the Crane-Yetter topological state
sum. That's its most important property.

Hope this helps.

Igor

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