#14775: Symmetric functions: extending Kronecker product, implementing Kronecker
product, extending antipode, extending forgotten basis, implementing Witt
basis
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Reporter: darij |
Owner: sage-combinat
Type: enhancement |
Status: needs_review
Priority: major |
Milestone: sage-5.11
Component: combinatorics |
Resolution:
Keywords: symmetric function, combinat, kronecker product, days49 |
Work issues:
Report Upstream: N/A |
Reviewers:
Authors: Darij Grinberg |
Merged in:
Dependencies: |
Stopgaps:
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Changes (by darij):
* status: needs_info => needs_review
Old description:
> Goals for this ticket:
>
> 1. The current version of itensor (the Kronecker product on the ring of
> symmetric function) only works when the ground ring is an algebra over
> the rationals. This is not a mathematically reasonable restriction. Fix
> this.
>
> 2. The Kronecker coproduct on the ring of symmetric function has to be
> implemented. Implement it.
>
> 3. The antipode on the ring of symmetric functions uses coercion into the
> powersum basis. This means it, too, is not getting computed over
> arbitrary base rings. Fix this.
>
> 4. The forgotten basis of Symm is defined by duality rather than by
> explicit formulas. Our duality methods use the powersum basis, again
> leading to errors for ground rings not being QQ-algebras.
>
> 5. The Witt symmetric functions form another basis of Symm. Implement
> them.
>
> 6. Implement Frobenius and Verschiebung operations on Symm without
> recourse to plethysm.
>
> The current file has the first goal achieved, but it uses some sly hacks.
> Can anyone tell me which of them are bad before I spread them into other
> functions (particularly for the second goal)?
New description:
This ticket does the following:
1. The current version of itensor (the Kronecker product on the ring of
symmetric function) only works when the ground ring is an algebra over the
rationals. This is not a mathematically reasonable restriction. Fix this.
2. The Kronecker coproduct on the ring of symmetric function has to be
implemented. Implement it.
3. The antipode on the ring of symmetric functions uses coercion into the
powersum basis. This means it, too, is not getting computed over arbitrary
base rings. Fix this.
4. The forgotten basis of Symm is defined by duality rather than by
explicit formulas. Our duality methods use the powersum basis, again
leading to errors for ground rings not being QQ-algebras.
5. The Witt symmetric functions form another basis of Symm. Implement
them.
6. Implement Frobenius and Verschiebung operations on Symm without
recourse to plethysm.
* Apply: [attachment:trac_14775-symmetric_functions_extended-dg.patch]
--
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14775#comment:14>
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