Trac #27036: FusionRing enhancements

The `FusionRing`, implemented in #26440 is the WZW fusion or Verlinde
algebra. It needs certain methods to be useful: the twist or ribbon
element, the S-matrix and the quantum dimension. This ticket is to
implement these methods.

The ticket #29615 is related to this one.

URL: https://trac.sagemath.org/27036
Reported by: bump
Ticket author(s): Guillermo Aboumrad, Daniel Bump
Reviewer(s): Travis Scrimshaw

src/doc/en/reference/combinat/module_list.rst

@@ -282,6 +282,7 @@ Comprehensive Module list
     sage/combinat/root_system/weight_lattice_realizations
     sage/combinat/root_system/weight_space
     sage/combinat/root_system/weyl_characters
+    sage/combinat/root_system/fusion_ring
     sage/combinat/root_system/weyl_group
     sage/combinat/rooted_tree
     sage/combinat/rsk

src/doc/en/reference/references/index.rst

@@ -347,6 +347,9 @@ REFERENCES:
 
 **B**
 
+.. [BaKi2001] Bakalov and Kirillov, *Lectures on tensor categories and modular functors*,
+            AMS (2001).
+
 .. [Ba1994] Kaushik Basu. *The Traveler's Dilemma: Paradoxes of
             Rationality in Game Theory*. The American Economic Review
             (1994): 391-395.
@@ -1984,6 +1987,9 @@ REFERENCES:
              1981, Pages 108-125, ISSN 0097-3165,:doi:`10.1016/0097-3165(81)90007-8`.
              (http://www.sciencedirect.com/science/article/pii/0097316581900078)
 
+.. [EGNO2015] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik,
+               *Tensor Categories*, AMS Mathematical Surveys and Monographs 205 (2015).
+
 .. [EGHLSVY] Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai
              Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina,
              "Introduction to representation theory",
@@ -4123,6 +4129,10 @@ REFERENCES:
 
 **N**
 
+.. [NaiRow2011] Naidu and Rowell, A finiteness property for braided fusion
+                categories. Algebr. Represent. Theory 14 (2011), no. 5, 837–855.
+                :arXiv:`0903.4157`.
+
 .. [Nas1950] John Nash. *Equilibrium points in n-person games.*
              Proceedings of the National Academy of Sciences 36.1
              (1950): 48-49.
@@ -4612,6 +4622,13 @@ REFERENCES:
 .. [Rot2006] Ron Roth, Introduction to Coding Theory, Cambridge
              University Press, 2006
 
+.. [Row2006] Eric Rowell, *From quantum groups to unitary modular tensor categories*.
+             In Representations of algebraic groups, quantum groups, and Lie algebras,
+             Contemp. Math., **413**, Amer. Math. Soc., Providence, RI, 2006.
+             :arXiv:`math/0503226`.
+
+.. [RoStWa2009] Eric Rowell, Richard Stong and Zhenghan Wang, *On classification
+        of modular tensor categories*, Comm. Math. Phys. 292, 343--389, 2009.
 
 .. [RR1997] Arun Ram and Jeffrey Remmel. *Applications of the Frobenius
             formulas and the characters of the symmetric group and the

src/sage/combinat/root_system/__init__.py

@@ -77,6 +77,7 @@
 ---------------------
 
 - :ref:`sage.combinat.root_system.weyl_characters`
+- :ref:`sage.combinat.root_system.fusion_ring`
 - :ref:`sage.combinat.root_system.integrable_representations`
 - :ref:`sage.combinat.root_system.branching_rules`
 - :ref:`sage.combinat.root_system.hecke_algebra_representation`

src/sage/combinat/root_system/all.py

@@ -19,8 +19,8 @@
             'ExtendedAffineWeylGroup')
 lazy_import('sage.combinat.root_system.coxeter_group', 'CoxeterGroup')
 lazy_import('sage.combinat.root_system.weyl_characters', ['WeylCharacterRing',
-                                                          'WeightRing',
-                                                          'FusionRing'])
+                                                          'WeightRing'])
+lazy_import('sage.combinat.root_system.fusion_ring', ['FusionRing'])
 from .branching_rules import BranchingRule, branching_rule_from_plethysm, branching_rule
 
 lazy_import('sage.combinat.root_system.non_symmetric_macdonald_polynomials', 'NonSymmetricMacdonaldPolynomials')