From d4a9857507537761d8430c5be7fd5e6af4e4500d Mon Sep 17 00:00:00 2001 From: Darij Grinberg Date: Mon, 3 Mar 2014 19:29:19 -0800 Subject: [PATCH] final doc changes --- src/sage/combinat/symmetric_group_algebra.py | 24 +++++++++++--------- 1 file changed, 13 insertions(+), 11 deletions(-) diff --git a/src/sage/combinat/symmetric_group_algebra.py b/src/sage/combinat/symmetric_group_algebra.py index 60e30055e20..b3df5f2d543 100644 --- a/src/sage/combinat/symmetric_group_algebra.py +++ b/src/sage/combinat/symmetric_group_algebra.py @@ -993,8 +993,8 @@ def seminormal_basis(self, mult='l2r'): where `f^{\lambda}` is the number of standard Young tableaux of shape `\lambda`. Note that `\kappa_{\lambda}` is an integer, - namely the product of all hook lengths of `\lambda` by the - hook length formula. In Sage, this integer can be computed by + namely the product of all hook lengths of `\lambda` (by the + hook length formula). In Sage, this integer can be computed by using :func:`sage.combinat.symmetric_group_algebra.kappa()`. Let `T` be a standard tableau. @@ -1036,9 +1036,9 @@ def seminormal_basis(self, mult='l2r'): e(T) \epsilon(\overline{T}). This element `\epsilon(T)` is implemented as - :func:`sage.combinat.symmetric_group_algebra.epsilon` for `R = \QQ`, - but it is also a particular case of the elements `\epsilon(T, S)` - defined below. + :func:`sage.combinat.symmetric_group_algebra.epsilon` for + `R = \QQ`, but it is also a particular case of the elements + `\epsilon(T, S)` defined below. Now let `S` be a further tableau of the same shape as `T` (possibly equal to `T`). Let `\pi_{T, S}` denote the @@ -1068,7 +1068,9 @@ def seminormal_basis(self, mult='l2r'): .. MATH:: - \epsilon(T, S) \epsilon(U, V) = \delta_{TV} \epsilon(U, S). + \epsilon(T, S) \epsilon(U, V) = \delta_{T, V} \epsilon(U, S) + + (where `\delta` stands for the Kronecker delta). .. WARNING:: @@ -1076,11 +1078,11 @@ def seminormal_basis(self, mult='l2r'): reverse of those given in some papers, for example [Ram1997]_. Using the other convention of multiplying permutations, we would instead have - `\epsilon(U, V) \epsilon(T, S) = \delta_{TV} \epsilon(U, S).` + `\epsilon(U, V) \epsilon(T, S) = \delta_{T, V} \epsilon(U, S).` - In other words, it consists of the matrix units in a - (particular) Artin-Wedderburn decomposition of `R S_n` into - a direct product of matrix algebras over `\QQ`. + In other words, Young's seminormal basis consists of the matrix + units in a (particular) Artin-Wedderburn decomposition of `R S_n` + into a direct product of matrix algebras over `\QQ`. The output of ``seminormal_basis`` is a list of all elements of the seminormal basis of ``self``. @@ -1107,7 +1109,7 @@ def seminormal_basis(self, mult='l2r'): .. [Ram1997] Arun Ram. *Seminormal representations of Weyl groups and Iwahori-Hecke algebras*. Proc. London Math. Soc. (3) - **75** (1997). 99-133. :arxiv:`math/9511223`. + **75** (1997). 99-133. :arxiv:`math/9511223v1`. http://www.ms.unimelb.edu.au/~ram/Publications/1997PLMSv75p99.pdf """ basis = []