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Translation for elements of a root lattice and related features and fixes #8811

Description

@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

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