From 6e224a479c810e48021ae678091ce1b9f3c4020c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fr=C3=A9d=C3=A9ric=20Chapoton?= Date: Thu, 17 Jun 2021 21:24:10 +0200 Subject: [PATCH] des typos dans le tuto fonctions symetriques --- tutorial-symmetric-functions.rst | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/tutorial-symmetric-functions.rst b/tutorial-symmetric-functions.rst index 4619e2f..df6f513 100644 --- a/tutorial-symmetric-functions.rst +++ b/tutorial-symmetric-functions.rst @@ -63,7 +63,7 @@ Abstract symmetric functions ---------------------------- 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`). -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`. +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`. - The **power sum** symmetric functions :math:`p_\mu=p_{\mu_1}p_{\mu_2}\cdots p_{\mu_2}` @@ -109,7 +109,7 @@ The keyword `verbose` allows you to make the injection quiet. sage: (q+t)*s[2,1,1] (q+t)*s[2, 1, 1] -Now that we have acces to all the bases we need, we can start to manipulate them. +Now that we have access to all the bases we need, we can start to manipulate them. Symmetric functions are indexed by partitions :math:`\mu`, with integers considered as partitions having size one (don't forget the brackets!):: @@ -286,7 +286,7 @@ in the variables, maybe written as a formal symmetric function in any chosen bas The ``pol`` input of the function ``from_polynomial(pol)`` is assumed to lie in a polynomial ring over the same base field as that used for the symmetric - functions, which thus has to be delared beforehand. + functions, which thus has to be declared beforehand. :: @@ -307,7 +307,7 @@ Finally, we can declare our polynomial and convert it into a symmetric function 2*m[1, 1, 1] + m[2, 1] -In the preceeding example, the base ring of polynomials is the same as the base +In the preceding example, the base ring of polynomials is the same as the base ring of symmetric polynomials considered, as checked by the following. :: @@ -385,7 +385,7 @@ For example, here we compute :math:`p_{22}+m_{11}s_{21}` in the elementary basis .. TOPIC:: Exercise - 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\}` + 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\}` 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``. @@ -428,7 +428,7 @@ http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/macdonald.ht Here are some examples involving the "combinatorial" Macdonald symmetric functions. These are eigenfunctions of the operator :math:`\nabla`. -(See below for more informations about :math:`\nabla`.) +(See below for more information about :math:`\nabla`.) :: @@ -749,7 +749,7 @@ of SAGE-variables to be considered as **constants**, using the option sage: p([2]).plethysm(g,exclude=[t]) p[2] + 1/3*t*p[2, 2, 2] + (-1/3*t)*p[6] -It is costumary to also write :math:`f[g]` for :math:`f\circ g` in +It is customary to also write :math:`f[g]` for :math:`f\circ g` in mathematical texts, but SAGE uses the shorthand notation :math:`f(g)` for better compatibility with python. For instance, the plethysm :math:`s_4\circ s_2`, may also be computed as @@ -1230,7 +1230,7 @@ For instance, we have :: 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[] -Skew Schur fonctions +Skew Schur functions ^^^^^^^^^^^^^^^^^^^^ arise when one considers the effect of coproduct on Schur functions themselves