Translation for elements of a root lattice and related features and fixes

[anneschilling]

This patch implements translation for elements of a root lattice and related features and fixes::

• New method translation for elements of a root lattice

• New methods reduced_word_of_alcove_morphism, dynkin_diagram_automorphism_of_alcove_morphism, reduced_word_of_translation, _test_reduced_word_of_translation

• Added extensive TESTS to compare with Kashiwara's private notes

• Fixed dynkin diagram for type BC

• Fixed CartanType.translation_factors to implement its specification correctly in affine type BC. Imported the full test suite from MuPAD-Combinat. Also fixes and tests for proper typing of the coefficients (ZZ/QQ).

• Bug fix in WeylGroup: the matrices were defined over QQ instead of the base ring of the underlying root lattice realization.

CC: @sagetrac-sage-combinat

Component: combinatorics

Keywords: root systems, affine weyl groups, translations

Author: Nicolas M. Thiéry

Reviewer: Anne Schilling

Merged: sage-4.4.4.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/8811

[nthiery]

Changed keywords from none to root systems, affine weyl groups, translations

[nthiery]

Reviewer: Anne Schilling, ...

[nthiery]

comment:4

Attachment: trac_8811_reduced_word_of_translations-nt.patch.gz

[anneschilling]

comment:5

I carefully checked the results of translation_factor, adding extra tests to compare with Kashiwara's private notes. Nicolas and I discussed at length the correct factors for type BC (see the corresponding doc in the code).

All tests passed on massena and all tests in combinat/root_systems passed on my machine.

Positive review.

[anneschilling]

Changed reviewer from Anne Schilling, ... to Anne Schilling

[mwhansen]

Merged: sage-4.4.4.alpha0

[mwhansen]

Changed author from Nicolas M. Thiery to Nicolas M. Thiéry