Implementation of the general Burau representation for Artin groups #39415
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Documentation preview for this PR (built with commit 02a7986; changes) is ready! 🎉 |
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I have started to look at it. A small comment first, there is an unused |
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Thanks for taking a look. As you can see in the doctest in |
src/sage/groups/artin.py
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| @@ -156,6 +164,132 @@ def coxeter_group_element(self, W=None): | |||
| In = W.index_set() | |||
| return W.prod(s[In[abs(i)-1]] for i in self.Tietze()) | |||
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| def burau_matrix(self, var='t', reduced=False): | |||
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I think reduced is not used.
| OUTPUT: | ||
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| The Burau matrix of the Artin group element over the Laurent | ||
| polynomial ring in the variable ``var``. |
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Maybe here it is worth to recall that for type A one gets unitary representation for braid groups.
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I added a comment about this in the main definition part.
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| if var == 't': | ||
| return ret | ||
| from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing |
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Would it be better to import it in the beginning of the file?
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Would it be better to import it in the beginning of the file?
I think local imports should be preferred as they perform better with regard to our modularization goals.
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Indeed, this is why I made them local. It's a bit of a judgement call I feel.
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LGTM |
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@enriqueartal Thank you. Can you approve the PR and then set the positive review label? |
| gens = [one - MS([SparseEntry(i, j, val(coxeter_matrix[index_set[i], index_set[j]])) | ||
| for j in range(n)]) | ||
| for i in range(n)] | ||
| invs = [one - q**-2 * MS([SparseEntry(i, j, val(coxeter_matrix[index_set[i], index_set[j]])) |
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Mathematically, there is no difference, but it would be more consistent with the citations from the literature if you swapped the indices in val(coxeter_matrix...). But if you don't see a need for that, that's fine with me.
…for Artin groups
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We provide an implementation of the generalized Burau representation
(following [this paper by Bapat and
Queffelec](https://arxiv.org/abs/2409.00144)). We also allow Artin
groups for arbitrary type to be constructed.
### 📝 Checklist
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preview.
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URL: sagemath#39415
Reported by: Travis Scrimshaw
Reviewer(s): Enrique Manuel Artal Bartolo, Sebastian Oehms, Travis Scrimshaw
We provide an implementation of the generalized Burau representation (following this paper by Bapat and Queffelec). We also allow Artin groups for arbitrary type to be constructed.
📝 Checklist
⌛ Dependencies