#15150: Implement NCSym
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Reporter: tscrim | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.12
Component: combinatorics | Resolution:
Keywords: | Merged in:
Authors: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: #15143 | Stopgaps:
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Comment (by zabrocki):
So about your question if `DNCSym` can be realized in the polynomial
algebra. The answer is 'probably' but no one has ever done this. Maybe
you can make a guess at the realization by what I will write. I will
guess that the dual to the complete basis is some sort of monomial basis
(the `w` basis is the dual to the `m` basis).
The morphism from `DNCSym` into `Sym` is 'the coefficient of `w_A` in
`h_\lambda` is equal to the coefficient of `m_\lambda` in `chi(m_A)`.'
Therefore,
{{{
\sum_{A} A^{!} w_A = h_\lambda
}}}
where the sum is over `A` with set partitions with part lengths a
partition `\lambda` and the `A^{!}` notation is from Rosas / Sagan p. 4.
--
Ticket URL: <http://trac.sagemath.org/ticket/15150#comment:23>
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