Timeline for Integer-matrix representation of a commutative ring
Current License: CC BY-SA 3.0
10 events
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| Dec 10, 2015 at 20:37 | comment | added | Yingfei Gu | This is certainly difficult, but in practical, it is possible to rule out many possibilities. For example, the matrix $N_i'$ have the same determinant of an integer matrix, therefore, $\prod_{j=1}^t \lambda_i^{(j)} \in \mathbb{Z}$ for all $i$. Maybe with other knowledge of integer matrix we can reduce the solutions further. | |
| Dec 10, 2015 at 20:31 | comment | added | Yingfei Gu | matrix $N_i$, and each eigenvalue produces a rep, so there are precisely $r=rank$ many 1-dimensional rep you can immediately get from data $N$. The next task is to figure out whether there is a way to put a subset, say pick $t<r$, of these 1-dimensional reps (most possibly some complex numbers) make diagonal matrices $N_i'=diag(\lambda_i^{(1)},\ldots,\lambda_i^{(t)})$, and find a basis transformation back to an integer matrix. | |
| Dec 10, 2015 at 20:24 | comment | added | Yingfei Gu | Hi Xiao-Gang, I don't have a full answer, but here is my thinking: Associativity of the ring ensure the matrix $\rm N_i$ itself is a representation. Question is whether it is reducible. Now if instead we work over field $k$, i.e., release the constrain of integer-matrix rep. According to Artin–Wedderburn theorem, the algebra here should be a direct sum of matrix algebra, and due to the commutativity you provide, these matrix algebra should commutes and therefore 1-dimensional. So as an algebra over field $k$, it is reducible, and its 1-dimensional rep can be achieved by diagonalizing | |
| Nov 18, 2015 at 12:38 | history | edited | Xiao-Gang Wen | CC BY-SA 3.0 |
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| Nov 17, 2015 at 17:49 | history | edited | Xiao-Gang Wen | CC BY-SA 3.0 |
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| Nov 17, 2015 at 17:44 | comment | added | Xiao-Gang Wen | Yes, I really want modules over this ring. Although I am also interested in module categories over the corresponding fusion category, but I do not have a list of fusion categories. I do have a list of fusion rings. So I like to start with fusion rings. Note that I asked for an algorithm to find the irreducible representations from the integer data $N^k_{ij}$. | |
| Nov 17, 2015 at 17:38 | comment | added | Qiaochu Yuan | Do you really want modules over this ring? I would guess that you would instead want module categories over the corresponding fusion category (assuming there is one). | |
| Nov 17, 2015 at 17:20 | history | edited | Xiao-Gang Wen | CC BY-SA 3.0 |
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| Nov 17, 2015 at 17:14 | history | edited | Xiao-Gang Wen | CC BY-SA 3.0 |
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| Nov 17, 2015 at 17:07 | history | asked | Xiao-Gang Wen | CC BY-SA 3.0 |