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Commit c870339

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Merge pull request #7 from fchapoton/des_typod
des typos dans le tuto fonctions symetriques
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‎tutorial-symmetric-functions.rst‎

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@@ -63,7 +63,7 @@ Abstract symmetric functions
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----------------------------
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We first describe how to manipulate "variable free" symmetric functions (with coefficients in the ring of rational coefficient fractions in :math:`q` and :math:`t`).
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Such functions are linear combinations of one of the six classical bases of symmetric functions; all indexed by interger partitions :math:`\mu=\mu_1\mu_2\cdots \mu_k`.
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Such functions are linear combinations of one of the six classical bases of symmetric functions; all indexed by integer partitions :math:`\mu=\mu_1\mu_2\cdots \mu_k`.
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- The **power sum** symmetric functions :math:`p_\mu=p_{\mu_1}p_{\mu_2}\cdots p_{\mu_2}`
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@@ -109,7 +109,7 @@ The keyword `verbose` allows you to make the injection quiet.
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sage: (q+t)*s[2,1,1]
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(q+t)*s[2, 1, 1]
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Now that we have acces to all the bases we need, we can start to manipulate them.
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Now that we have access to all the bases we need, we can start to manipulate them.
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Symmetric functions are indexed by partitions :math:`\mu`, with integers considered
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as partitions having size one (don't forget the brackets!)::
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The ``pol`` input of the function ``from_polynomial(pol)`` is assumed to
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lie in a polynomial ring over the same base field as that used for the symmetric
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functions, which thus has to be delared beforehand.
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functions, which thus has to be declared beforehand.
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::
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2*m[1, 1, 1] + m[2, 1]
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In the preceeding example, the base ring of polynomials is the same as the base
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In the preceding example, the base ring of polynomials is the same as the base
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ring of symmetric polynomials considered, as checked by the following.
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::
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.. TOPIC:: Exercise
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It is well konwn that :math:`h_n(X) = \sum \limits_{\mu \vdash n} \dfrac{p_{\mu}(x)}{z_{\mu}}`. Verify this result for :math:`n \in \{1,2,3,4\}`
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It is well known that :math:`h_n(X) = \sum \limits_{\mu \vdash n} \dfrac{p_{\mu}(x)}{z_{\mu}}`. Verify this result for :math:`n \in \{1,2,3,4\}`
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Note that there exists a function ``zee()`` which takes a partition :math:`\mu` and gives back the value of :math:`z_{\mu}`. To use this function, you should import it from* ``sage.combinat.sf.sfa``.
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@@ -428,7 +428,7 @@ http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/macdonald.ht
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Here are some examples involving the "combinatorial" Macdonald symmetric functions.
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These are eigenfunctions of the operator :math:`\nabla`.
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(See below for more informations about :math:`\nabla`.)
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(See below for more information about :math:`\nabla`.)
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::
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sage: p([2]).plethysm(g,exclude=[t])
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p[2] + 1/3*t*p[2, 2, 2] + (-1/3*t)*p[6]
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It is costumary to also write :math:`f[g]` for :math:`f\circ g` in
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It is customary to also write :math:`f[g]` for :math:`f\circ g` in
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mathematical texts, but SAGE uses the shorthand notation :math:`f(g)`
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for better compatibility with python. For instance, the plethysm
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:math:`s_4\circ s_2`, may also be computed as
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s[] # s[3, 2, 1] + s[1] # s[2, 2, 1] + s[1] # s[3, 1, 1] + s[1] # s[3, 2] + s[1, 1] # s[2, 1, 1] + s[1, 1] # s[2, 2] + s[1, 1] # s[3, 1] + s[1, 1, 1] # s[2, 1] + s[2] # s[2, 1, 1] + s[2] # s[2, 2] + s[2] # s[3, 1] + s[2, 1] # s[1, 1, 1] + 2*s[2, 1] # s[2, 1] + s[2, 1] # s[3] + s[2, 1, 1] # s[1, 1] + s[2, 1, 1] # s[2] + s[2, 2] # s[1, 1] + s[2, 2] # s[2] + s[2, 2, 1] # s[1] + s[3] # s[2, 1] + s[3, 1] # s[1, 1] + s[3, 1] # s[2] + s[3, 1, 1] # s[1] + s[3, 2] # s[1] + s[3, 2, 1] # s[]
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Skew Schur fonctions
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Skew Schur functions
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^^^^^^^^^^^^^^^^^^^^
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arise when one considers the effect of coproduct on Schur functions themselves

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