Compute dimensions of simple symmetric group modules over positive characteristic #36724
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fchapoton
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Nov 16, 2023
fchapoton
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Nov 16, 2023
fchapoton
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Nov 16, 2023
fchapoton
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Nov 16, 2023
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…ules over positive characteristics.
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<!-- ^^^^^ Please provide a concise, informative and self-explanatory title. Don't put issue numbers in there, do this in the PR body below. For example, instead of "Fixes sagemath#1234" use "Introduce new method to calculate 1+1" --> <!-- Describe your changes here in detail --> This provides an implementation of the Specht modules over arbitrary fields in the standard tableaux basis. To do so, we also implement the tabloid module, where the tabloids are implemented as `OrderedSetsPartitions`. Using this, we can construct the maximal submodule and simple submodule from a Specht module. We also implement a method to compute the Brauer character of the representations. <!-- Why is this change required? What problem does it solve? --> <!-- If this PR resolves an open issue, please link to it here. For example "Fixes sagemath#12345". --> <!-- If your change requires a documentation PR, please link it appropriately. --> ### 📝 Checklist <!-- Put an `x` in all the boxes that apply. --> <!-- If your change requires a documentation PR, please link it appropriately --> <!-- If you're unsure about any of these, don't hesitate to ask. We're here to help! --> <!-- Feel free to remove irrelevant items. --> - [x] The title is concise, informative, and self-explanatory. - [x] The description explains in detail what this PR is about. - [x] I have linked a relevant issue or discussion. - [x] I have created tests covering the changes. - [x] I have updated the documentation accordingly. ### ⌛ Dependencies <!-- List all open PRs that this PR logically depends on - sagemath#12345: short description why this is a dependency - sagemath#34567: ... --> - sagemath#36724: Trivial conflict with working on the same files. - sagemath#36985: So the 0 dimensional tests will pass. <!-- If you're unsure about any of these, don't hesitate to ask. We're here to help! --> URL: sagemath#36748 Reported by: Travis Scrimshaw Reviewer(s): Darij Grinberg, Frédéric Chapoton, Martin Rubey, Travis Scrimshaw
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We can compute Specht module dimensions, but an interesting open problem is to compute the dimensions of simple modules for
S_nover fields of positive characteristic. We provide a way to compute dimensions by using the pullback of the polytabloid inner product and computing the rank of the Gram matrix.This is based off of code provided by Jackson Walters.
We also implement a more efficient column stabilizer method for standard tableaux.
📝 Checklist
⌛ Dependencies