#15408: corrections and improvements to `inner_plethysm` method in symmetric
functions
-------------------------------------+-------------------------------------
Reporter: zabrocki | Owner:
Type: defect | Status: needs_review
Priority: major | Milestone: sage-6.0
Component: combinatorics | Resolution:
Keywords: symmertic | Merged in:
functions, inner plethysm | Reviewers:
Authors: Mike Zabrocki | Work issues:
Report Upstream: N/A | Commit:
Branch: | 3c4ce5b08218532b6c1f72d552dc7ad5758780d6
public/combinat/15408/zabrocki/inner_plethysm| Stopgaps:
Dependencies: |
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Comment (by saliola):
Here are two slight rewordings of what Mike proposes.
1. Change:
{{{
Here is an axiomatic definition of the operation (where in the equations
below we denote outer product
(:meth:`~sage.categories.algebras_with_basis.AlgebrasWithBasis.ParentMethods.product`)
by `\cdot` and the Kronecker product (:meth:`itensor`) by `\ast`):
}}}
to
{{{
First we describe the axiomatic definition of the operation; see below for
the representation-theoretic definition.
In the following equations, we denote outer product
(:meth:`~sage.categories.algebras_with_basis.AlgebrasWithBasis.ParentMethods.product`)
by `\cdot` and the Kronecker product (:meth:`itensor`) by `\ast`).
}}}
2. For the representation-theoretic definition:
{{{
This operation admits a representation-theoretic interpretation
in the case where `f` is a Schur function `s_\lambda` and
`g` is a homogeneous degree `n` symmetric function with
positive integral coefficients in the Schur basis.
The symmetric function ``f.inner_plethysm(g)`` is the Frobenius
image of the `S_n`-representation constructed as follows.
The assumptions on `g` imply that `g` is the Frobenius image of a
representation `\rho` of the symmetric group `S_n`:
.. MATH::
\rho : S_n \to GL_N
If the degree `N` of this representation is greater than or equal
to the number of parts of `\lambda`, then `f`, which denotes `s_\lambda`,
corresponds to the character of some irreducible `GL_N`-representation,
say
.. MATH::
\sigma : GL_N \to GL_M
The composition `\sigma \circ \rho : S_n \to GL_M` is a representation
of `S_n` whose Frobenius image is precisely ``f.inner_plethysm(g)``.
If `N` is less than the number of parts of `\lambda`,
then ``f.inner_plethysm(g)`` is `0` by definition.
}}}
Personally, I like to see `\rho: S_n \to GL_N` and `\sigma: GL_N \to
GL_M` displayed in this way with domain and codomain, so that I can see
the composition. So I re-worded the text to take this into account.
Otherwise, patch looks good to me.
----
New commits:
||[[http://git.sagemath.org/sage.git/commit/?id=3c4ce5b|3c4ce5b]]||some
doc changes II, for some reason||
||[[http://git.sagemath.org/sage.git/commit/?id=13b9b6d|13b9b6d]]||docstrings
reviewed||
||[[http://git.sagemath.org/sage.git/commit/?id=6302b95|6302b95]]||some
doc changes||
--
Ticket URL: <http://trac.sagemath.org/ticket/15408#comment:14>
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