How to represent SU(N) as polynomials of the irreducible representations of SU(2) in complex vector space of...












0














Suppose $S^x$,$S^y$ and $S^z$ are the irreducible representations of $SU(2)$ in a complex vector space, V, of dimension N. By Burnside's theorem, $I$, $S^x$,$S^y$ and $S^z$ for a basis in the ring of endomorphims of V.



Now take $T_k$ as the generators of $SU(N)$. By the definition of $SU(N)$, $T_k$ is a subset of endomorphims in V.



Thus, I expect I should be able to write:



$
T_k=f_k(S^x,S^y,S^z,I)
$



with $f_k$ being a polynomial.



Is this expecation correct. Did I understand Burnside's theorem and the definition of SU(N) correctly?




  • If so, what can I say about $f_k$?

  • What is the highest degree polynomial required?

  • Do I need products like $S^xS^y$ or do monomials in $(S^i)^n$ suffice










share|cite|improve this question






















  • What "Burnside's theorem" are you referring to?
    – Qiaochu Yuan
    Nov 21 '18 at 1:01










  • I didn't know there were multiple: it says that if a moniod of linear transformations acts irreducibly then it is a basis for the ring of endomorphisms, I found it in this paper: arxiv.org/abs/cond-mat/0207106
    – Shane P Kelly
    Nov 21 '18 at 1:07






  • 2




    I don't know this result by that name; for me this is a special case of the Jacobson density theorem (en.wikipedia.org/wiki/Jacobson_density_theorem). In mathematics "Burnside's theorem" refers to a theorem about finite groups: en.wikipedia.org/wiki/Burnside_theorem
    – Qiaochu Yuan
    Nov 21 '18 at 1:18








  • 2




    Your language is somewhat imprecise but you are basically right, if by "polynomial" you mean "noncommutative polynomial" (and "basis" in your second sentence should be "set of generators"). I don't know that there's anything nice to say about the $f_k$ and I would not enjoy attempting to calculate them. You need to make many noncanonical choices to define $f_k$, for example a choice of basis of $mathfrak{su}(2)$, a choice of inner product on $V$, and a choice of basis of $mathfrak{su}(V)$.
    – Qiaochu Yuan
    Nov 21 '18 at 1:20












  • @QiaochuYuan Regarding the two Burnside theorems: the lesser known one does really go by that name too, a great proof is prestented here: core.ac.uk/download/pdf/82680953.pdf
    – Vincent
    Nov 21 '18 at 14:16
















0














Suppose $S^x$,$S^y$ and $S^z$ are the irreducible representations of $SU(2)$ in a complex vector space, V, of dimension N. By Burnside's theorem, $I$, $S^x$,$S^y$ and $S^z$ for a basis in the ring of endomorphims of V.



Now take $T_k$ as the generators of $SU(N)$. By the definition of $SU(N)$, $T_k$ is a subset of endomorphims in V.



Thus, I expect I should be able to write:



$
T_k=f_k(S^x,S^y,S^z,I)
$



with $f_k$ being a polynomial.



Is this expecation correct. Did I understand Burnside's theorem and the definition of SU(N) correctly?




  • If so, what can I say about $f_k$?

  • What is the highest degree polynomial required?

  • Do I need products like $S^xS^y$ or do monomials in $(S^i)^n$ suffice










share|cite|improve this question






















  • What "Burnside's theorem" are you referring to?
    – Qiaochu Yuan
    Nov 21 '18 at 1:01










  • I didn't know there were multiple: it says that if a moniod of linear transformations acts irreducibly then it is a basis for the ring of endomorphisms, I found it in this paper: arxiv.org/abs/cond-mat/0207106
    – Shane P Kelly
    Nov 21 '18 at 1:07






  • 2




    I don't know this result by that name; for me this is a special case of the Jacobson density theorem (en.wikipedia.org/wiki/Jacobson_density_theorem). In mathematics "Burnside's theorem" refers to a theorem about finite groups: en.wikipedia.org/wiki/Burnside_theorem
    – Qiaochu Yuan
    Nov 21 '18 at 1:18








  • 2




    Your language is somewhat imprecise but you are basically right, if by "polynomial" you mean "noncommutative polynomial" (and "basis" in your second sentence should be "set of generators"). I don't know that there's anything nice to say about the $f_k$ and I would not enjoy attempting to calculate them. You need to make many noncanonical choices to define $f_k$, for example a choice of basis of $mathfrak{su}(2)$, a choice of inner product on $V$, and a choice of basis of $mathfrak{su}(V)$.
    – Qiaochu Yuan
    Nov 21 '18 at 1:20












  • @QiaochuYuan Regarding the two Burnside theorems: the lesser known one does really go by that name too, a great proof is prestented here: core.ac.uk/download/pdf/82680953.pdf
    – Vincent
    Nov 21 '18 at 14:16














0












0








0







Suppose $S^x$,$S^y$ and $S^z$ are the irreducible representations of $SU(2)$ in a complex vector space, V, of dimension N. By Burnside's theorem, $I$, $S^x$,$S^y$ and $S^z$ for a basis in the ring of endomorphims of V.



Now take $T_k$ as the generators of $SU(N)$. By the definition of $SU(N)$, $T_k$ is a subset of endomorphims in V.



Thus, I expect I should be able to write:



$
T_k=f_k(S^x,S^y,S^z,I)
$



with $f_k$ being a polynomial.



Is this expecation correct. Did I understand Burnside's theorem and the definition of SU(N) correctly?




  • If so, what can I say about $f_k$?

  • What is the highest degree polynomial required?

  • Do I need products like $S^xS^y$ or do monomials in $(S^i)^n$ suffice










share|cite|improve this question













Suppose $S^x$,$S^y$ and $S^z$ are the irreducible representations of $SU(2)$ in a complex vector space, V, of dimension N. By Burnside's theorem, $I$, $S^x$,$S^y$ and $S^z$ for a basis in the ring of endomorphims of V.



Now take $T_k$ as the generators of $SU(N)$. By the definition of $SU(N)$, $T_k$ is a subset of endomorphims in V.



Thus, I expect I should be able to write:



$
T_k=f_k(S^x,S^y,S^z,I)
$



with $f_k$ being a polynomial.



Is this expecation correct. Did I understand Burnside's theorem and the definition of SU(N) correctly?




  • If so, what can I say about $f_k$?

  • What is the highest degree polynomial required?

  • Do I need products like $S^xS^y$ or do monomials in $(S^i)^n$ suffice







group-theory representation-theory lie-algebras






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share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 21 '18 at 0:48









Shane P Kelly

1106




1106












  • What "Burnside's theorem" are you referring to?
    – Qiaochu Yuan
    Nov 21 '18 at 1:01










  • I didn't know there were multiple: it says that if a moniod of linear transformations acts irreducibly then it is a basis for the ring of endomorphisms, I found it in this paper: arxiv.org/abs/cond-mat/0207106
    – Shane P Kelly
    Nov 21 '18 at 1:07






  • 2




    I don't know this result by that name; for me this is a special case of the Jacobson density theorem (en.wikipedia.org/wiki/Jacobson_density_theorem). In mathematics "Burnside's theorem" refers to a theorem about finite groups: en.wikipedia.org/wiki/Burnside_theorem
    – Qiaochu Yuan
    Nov 21 '18 at 1:18








  • 2




    Your language is somewhat imprecise but you are basically right, if by "polynomial" you mean "noncommutative polynomial" (and "basis" in your second sentence should be "set of generators"). I don't know that there's anything nice to say about the $f_k$ and I would not enjoy attempting to calculate them. You need to make many noncanonical choices to define $f_k$, for example a choice of basis of $mathfrak{su}(2)$, a choice of inner product on $V$, and a choice of basis of $mathfrak{su}(V)$.
    – Qiaochu Yuan
    Nov 21 '18 at 1:20












  • @QiaochuYuan Regarding the two Burnside theorems: the lesser known one does really go by that name too, a great proof is prestented here: core.ac.uk/download/pdf/82680953.pdf
    – Vincent
    Nov 21 '18 at 14:16


















  • What "Burnside's theorem" are you referring to?
    – Qiaochu Yuan
    Nov 21 '18 at 1:01










  • I didn't know there were multiple: it says that if a moniod of linear transformations acts irreducibly then it is a basis for the ring of endomorphisms, I found it in this paper: arxiv.org/abs/cond-mat/0207106
    – Shane P Kelly
    Nov 21 '18 at 1:07






  • 2




    I don't know this result by that name; for me this is a special case of the Jacobson density theorem (en.wikipedia.org/wiki/Jacobson_density_theorem). In mathematics "Burnside's theorem" refers to a theorem about finite groups: en.wikipedia.org/wiki/Burnside_theorem
    – Qiaochu Yuan
    Nov 21 '18 at 1:18








  • 2




    Your language is somewhat imprecise but you are basically right, if by "polynomial" you mean "noncommutative polynomial" (and "basis" in your second sentence should be "set of generators"). I don't know that there's anything nice to say about the $f_k$ and I would not enjoy attempting to calculate them. You need to make many noncanonical choices to define $f_k$, for example a choice of basis of $mathfrak{su}(2)$, a choice of inner product on $V$, and a choice of basis of $mathfrak{su}(V)$.
    – Qiaochu Yuan
    Nov 21 '18 at 1:20












  • @QiaochuYuan Regarding the two Burnside theorems: the lesser known one does really go by that name too, a great proof is prestented here: core.ac.uk/download/pdf/82680953.pdf
    – Vincent
    Nov 21 '18 at 14:16
















What "Burnside's theorem" are you referring to?
– Qiaochu Yuan
Nov 21 '18 at 1:01




What "Burnside's theorem" are you referring to?
– Qiaochu Yuan
Nov 21 '18 at 1:01












I didn't know there were multiple: it says that if a moniod of linear transformations acts irreducibly then it is a basis for the ring of endomorphisms, I found it in this paper: arxiv.org/abs/cond-mat/0207106
– Shane P Kelly
Nov 21 '18 at 1:07




I didn't know there were multiple: it says that if a moniod of linear transformations acts irreducibly then it is a basis for the ring of endomorphisms, I found it in this paper: arxiv.org/abs/cond-mat/0207106
– Shane P Kelly
Nov 21 '18 at 1:07




2




2




I don't know this result by that name; for me this is a special case of the Jacobson density theorem (en.wikipedia.org/wiki/Jacobson_density_theorem). In mathematics "Burnside's theorem" refers to a theorem about finite groups: en.wikipedia.org/wiki/Burnside_theorem
– Qiaochu Yuan
Nov 21 '18 at 1:18






I don't know this result by that name; for me this is a special case of the Jacobson density theorem (en.wikipedia.org/wiki/Jacobson_density_theorem). In mathematics "Burnside's theorem" refers to a theorem about finite groups: en.wikipedia.org/wiki/Burnside_theorem
– Qiaochu Yuan
Nov 21 '18 at 1:18






2




2




Your language is somewhat imprecise but you are basically right, if by "polynomial" you mean "noncommutative polynomial" (and "basis" in your second sentence should be "set of generators"). I don't know that there's anything nice to say about the $f_k$ and I would not enjoy attempting to calculate them. You need to make many noncanonical choices to define $f_k$, for example a choice of basis of $mathfrak{su}(2)$, a choice of inner product on $V$, and a choice of basis of $mathfrak{su}(V)$.
– Qiaochu Yuan
Nov 21 '18 at 1:20






Your language is somewhat imprecise but you are basically right, if by "polynomial" you mean "noncommutative polynomial" (and "basis" in your second sentence should be "set of generators"). I don't know that there's anything nice to say about the $f_k$ and I would not enjoy attempting to calculate them. You need to make many noncanonical choices to define $f_k$, for example a choice of basis of $mathfrak{su}(2)$, a choice of inner product on $V$, and a choice of basis of $mathfrak{su}(V)$.
– Qiaochu Yuan
Nov 21 '18 at 1:20














@QiaochuYuan Regarding the two Burnside theorems: the lesser known one does really go by that name too, a great proof is prestented here: core.ac.uk/download/pdf/82680953.pdf
– Vincent
Nov 21 '18 at 14:16




@QiaochuYuan Regarding the two Burnside theorems: the lesser known one does really go by that name too, a great proof is prestented here: core.ac.uk/download/pdf/82680953.pdf
– Vincent
Nov 21 '18 at 14:16










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