# 15-j sybmols

by T.M.Tlas@gmail.com
Tags: sybmols
 P: n/a 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 0 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 15-j ones. Can anybody point to a reference which does, and ARE there similar identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..) Thank you for your time, Tim
 P: n/a > 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 15-j ones. > Can anybody point to a reference which does, and ARE there similar > identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..) > > Thank you for your time, > Tim 3j -> coupling of two angular momentum 6j -> coupling of three angular momentum 9j -> 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
 P: n/a > 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 15-j ones. > Can anybody point to a reference which does, and ARE there similar > identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..) > > Thank you for your time, > Tim 3j -> coupling of two angular momentum 6j -> coupling of three angular momentum 9j -> 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
P: n/a

## 15-j sybmols

<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 15-j ones.
> Can anybody point to a reference which does, and ARE there similar
> identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..)
>
> Thank you for your time,
> Tim

3j -> coupling of two angular momentum
6j -> coupling of three angular momentum
9j -> 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

 P: n/a > 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 15-j ones. > Can anybody point to a reference which does, and ARE there similar > identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..) > > Thank you for your time, > Tim 3j -> coupling of two angular momentum 6j -> coupling of three angular momentum 9j -> 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
 P: n/a > 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 15-j ones. > Can anybody point to a reference which does, and ARE there similar > identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..) > > Thank you for your time, > Tim 3j -> coupling of two angular momentum 6j -> coupling of three angular momentum 9j -> 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
 P: n/a > 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 15-j ones. > Can anybody point to a reference which does, and ARE there similar > identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..) > > Thank you for your time, > Tim 3j -> coupling of two angular momentum 6j -> coupling of three angular momentum 9j -> 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
 P: n/a > 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 15-j ones. > Can anybody point to a reference which does, and ARE there similar > identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..) > > Thank you for your time, > Tim 3j -> coupling of two angular momentum 6j -> coupling of three angular momentum 9j -> 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
 P: n/a > 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 15-j ones. > Can anybody point to a reference which does, and ARE there similar > identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..) > > Thank you for your time, > Tim 3j -> coupling of two angular momentum 6j -> coupling of three angular momentum 9j -> 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
 P: n/a > 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 15-j ones. > Can anybody point to a reference which does, and ARE there similar > identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..) > > Thank you for your time, > Tim 3j -> coupling of two angular momentum 6j -> coupling of three angular momentum 9j -> 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
 P: n/a 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 (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 0 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 (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 15-j > ones. Can anybody point to a reference which does, and ARE there > similar identities in the 15-j 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 \ / \ / \____/ as linear combinations of \ / . / \ | / \ | / \ / \ 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
 P: n/a 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 (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 0 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 (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 15-j > ones. Can anybody point to a reference which does, and ARE there > similar identities in the 15-j 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 \ / \ / \____/ as linear combinations of \ / . / \ | / \ | / \ / \ 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
 P: n/a 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 (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 0 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 (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 15-j > ones. Can anybody point to a reference which does, and ARE there > similar identities in the 15-j 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 \ / \ / \____/ as linear combinations of \ / . / \ | / \ | / \ / \ 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
 P: n/a 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 (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 0 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 (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 15-j > ones. Can anybody point to a reference which does, and ARE there > similar identities in the 15-j 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 \ / \ / \____/ as linear combinations of \ / . / \ | / \ | / \ / \ 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
 P: n/a 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 (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 0 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 (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 15-j > ones. Can anybody point to a reference which does, and ARE there > similar identities in the 15-j 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 \ / \ / \____/ as linear combinations of \ / . / \ | / \ | / \ / \ 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
 P: n/a 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 (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 0 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 (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 15-j > ones. Can anybody point to a reference which does, and ARE there > similar identities in the 15-j 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 \ / \ / \____/ as linear combinations of \ / . / \ | / \ | / \ / \ 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
 P: n/a 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 (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 0 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 (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 15-j > ones. Can anybody point to a reference which does, and ARE there > similar identities in the 15-j 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 \ / \ / \____/ as linear combinations of \ / . / \ | / \ | / \ / \ 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
 P: n/a 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 (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 0 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 (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 15-j > ones. Can anybody point to a reference which does, and ARE there > similar identities in the 15-j 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 \ / \ / \____/ as linear combinations of \ / . / \ | / \ | / \ / \ 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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