Timeline for A decomposition of the "spin representation" of SL(2)
Current License: CC BY-SA 2.5
15 events
| when toggle format | what | by | license | comment | |
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| Feb 26, 2011 at 7:35 | vote | accept | Leonid Petrov | ||
| Feb 25, 2011 at 18:40 | answer | added | ARupinski | timeline score: 11 | |
| Oct 20, 2010 at 5:48 | comment | added | Leonid Petrov | José: Thank you for the explanation about the signature of the inner product. I am indeed interested in this representation (denote it by $T_k$) of $G=SL(2,R)$ of dimension $2^k$ that you just explained above. However, I have a candidate for the character of $T_k$. The representation $T_k$ acts in the space with basis $e_{j_1}\wedge\dots\wedge e_{j_l}$, where $1\le j_1<\dots<j_l\le k$, and for the diagonal generator $H$ of the Lie algebra $sl_2$ we have $T_k(H)e_{j_1}\wedge\dots\wedge e_{j_l} = (2(j_1+\dots+j_l)-k(k+1)/2)e_{j_1}\wedge\dots\wedge e_{j_l}$, which defines the character. | |
| Oct 20, 2010 at 1:19 | answer | added | Abdelmalek Abdesselam | timeline score: 3 | |
| Oct 19, 2010 at 21:09 | comment | added | José Figueroa-O'Farrill | (cont'd) From your response to Jim's comment, though, it seems that you are interested in a different representation: namely, the one afforded by the exterior algebra of $V$. Hence my original confusion. I guess it's the use of the word "spin" which confuses me. | |
| Oct 19, 2010 at 21:05 | comment | added | José Figueroa-O'Farrill | Leonid: Let $G=\operatorname{SL}(2,\mathbb{R})$ and let $W$ be thedefining representation of $G$: it is real and 2-dimensional. The $N$-dimensional representation $V$ of $G$ is the ($N-1$)-fold symmetric tensor power of $W$: $V = \operatorname{Sym}^{N-1} W$. If $N=2k+1$, $V$ has a $G$-invariant inner product of signature $(k,k+1)$. The group $\operatorname{Spin}(k,k+1)$ has a unique irreducible spinor representation: it is real and of dimension $2^k$. The title of your question suggests that it is the restriction to $G$ of that representation that you are interested in. | |
| Oct 19, 2010 at 19:58 | comment | added | Leonid Petrov | Jim: I try to identify a certain (other) representation of SL(2,R) which has dimension $2^N$. Well, I know its decomposition into irreducibles, but I do not know which type of "familiar" reps it is. This other representation (in the exterior algebra $\wedge V$) has the following action of the $sl_2$ diagonal generator: $H(e_{i_1}\wedge\dots\wedge e_{i_k})=(2(i_1+…+i_k)−N(N+1)/2)∗(e_{i_1}\wedge\dots\wedge e_{i_k})$. The "spin" representation that is in the question is just a candidate to be equal to this other one. | |
| Oct 19, 2010 at 19:50 | comment | added | Leonid Petrov | José: spinor module over what do you mean? I mean the representation of $SL(2,R)$: take the spin representation of $O(V,b)$ and restrict it to $SL(2,R)$ - you get a $2^N$-dimensional representation of $SL(2,R)$. | |
| Oct 19, 2010 at 18:15 | comment | added | José Figueroa-O'Farrill | I am a little confused about what is meant by the spin representation in this question. If the dimension of $V$ is $N=2k+1$, then the dimension of the irreducible spinor module $\Sigma V$ is $2^k$. It is the Clifford algebra which has dimension $2^N$ but as a Spin module this is just the exterior algebra $\Lambda V$. So which representation is the question about? $\Lambda V$ or $\Sigma V$? | |
| Oct 19, 2010 at 16:37 | comment | added | Melania | Try use characters. | |
| Oct 19, 2010 at 16:03 | comment | added | Jim Humphreys | I'm not sure whether there is a nice answer to this question, since random restrictions to rank 1 subgroups get unreasonably complicated. But in any case it might be helpful to know something about what motivates the question. | |
| Oct 19, 2010 at 15:19 | history | edited | Leonid Petrov | CC BY-SA 2.5 |
added 13 characters in body
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| Oct 19, 2010 at 15:07 | history | undeleted | Leonid Petrov | ||
| Oct 19, 2010 at 15:07 | history | deleted | Leonid Petrov | ||
| Oct 19, 2010 at 15:07 | history | asked | Leonid Petrov | CC BY-SA 2.5 |