Trac #19594: Implement the cactus group

We implement the cactus group ''J'',,n,, (of type A). The cactus group
has important applications in category and representation theory.

This group is not available in GAP as far as I can tell.

URL: https://trac.sagemath.org/19594
Reported by: tscrim
Ticket author(s): Travis Scrimshaw
Reviewer(s): Dima Pasechnik

Author: Release Manager

build/pkgs/configure/checksums.ini

@@ -1,4 +1,4 @@
 tarball=configure-VERSION.tar.gz
-sha1=a22c60b5adf1e70fc414df24a44fa390fe826b01
-md5=61556448c0562408a8b6f61f22d18692
-cksum=2169543810
+sha1=e95336247a7b690d91bc95af9031cd94f4808760
+md5=35b7a83c04310cba28e5c078443ec35e
+cksum=2310783026

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-104dde9eaa8231e4f0755c6855c1eabb43239dec
+1964374b1152cf0e5ac3c811f06db7de95572da2

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

@@ -19,11 +19,13 @@ Groups
    sage/groups/cubic_braid
    sage/groups/indexed_free_group
    sage/groups/raag
+   sage/groups/cactus_group
    sage/groups/group_exp
    sage/groups/group_semidirect_product
    sage/groups/misc_gps/misc_groups
    sage/groups/semimonomial_transformations/semimonomial_transformation_group
    sage/groups/semimonomial_transformations/semimonomial_transformation
+   sage/groups/kernel_subgroup
    sage/groups/class_function
    sage/groups/conjugacy_classes
 

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

@@ -450,6 +450,10 @@ REFERENCES:
 .. [BCDM2019] \T. Beyne, Y. L. Chen, C. Dobraunig, B. Mennink. *Elephant v1* (2019)
               https://csrc.nist.gov/CSRC/media/Projects/Lightweight-Cryptography/documents/round-1/spec-doc/elephant-spec.pdf
 
+.. [BCL2022] Paolo Bellingeri, Hugo Chemin, and Victoria Lebed.
+             *Cactus groups, twin groups, and right-angled Artin groups*.
+             Preprint, :arxiv:`2209.08813` (2022).
+
 .. [BeBo2009] Olivier Bernardi and Nicolas Bonichon, *Intervals in Catalan
               lattices and realizers of triangulations*, JCTA 116 (2009)
 
@@ -1875,6 +1879,11 @@ REFERENCES:
              *Introduction to Lattices and Order*,
              Cambridge University Press, 1997.
 
+.. [DJS2003] \M. Davis, T. Januszkiewicz, and R. Scott.
+             *Fundamental groups of blow-ups*. Selecta Math.,
+             Adv. Math. ***177** no. 1 (2002) pp. 115-179.
+             :arxiv:`math/0203127`.
+
 .. [DB1996] K. Duggal, A. Bejancu,
             *Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications*,
             Mathematics and Its Applications, 1996.
@@ -4480,6 +4489,11 @@ REFERENCES:
              of graphs*, Linear Algebra and its Applications 103 (1988),
              119–131.
 
+.. [Most2019] Jacob Mostovoy.
+              *The pure cactus group is residually nilpotent*.
+              Archiv der Math., **113** (2019). pp. 229-235.
+              :arxiv:`1804.09165`.
+
 .. [MNO1994] Alexander Molev, Maxim Nazarov, and Grigori Olshanski.
              *Yangians and classical Lie algebras*. (1994)
              :arxiv:`hep-th/9409025`
@@ -6099,6 +6113,11 @@ REFERENCES:
              pages 150--168, 1932,
              `available on JSTOR <http://www.jstor.org/stable/2371086>`_
 
+.. [White2015] Noah White.
+               *The monodromy of real Bethe vectors for the Gaudin model*.
+               J. Combin. Algebra, **2** no. 3 (2018). pp. 259-300.
+               :arxiv:`1511.04740`.
+
 .. [Wich1997] Tim Wichmann. Der FGLM Algorithmus - verallgemeinert und implementiert in Singular
               Diploma Thesis (University of Kaiserslautern), 1997.
 
@@ -6213,6 +6232,9 @@ REFERENCES:
 .. [Yu2007] \K. Yu, *p-adic logarithmic forms and group varieties. III.*
             Forum Math., 19(2):187–280, 2007.
 
+.. [Yu2022] Runze Yu. *linearity of generalized cactus groups*.
+            Preprint, :arxiv:`2202.00860` (2022).
+
 .. [Yun1976] Yun, David YY. On square-free decomposition
              algorithms. In Proceedings of the third ACM symposium on
              Symbolic and algebraic computation, pp. 26-35. ACM, 1976.

src/sage/combinat/permutation.py

@@ -6817,6 +6817,11 @@ def _coerce_map_from_(self, G):
             True
             sage: P.has_coerce_map_from(Permutations(7))
             False
+
+            sage: P.has_coerce_map_from(groups.misc.Cactus(5))
+            True
+            sage: P.has_coerce_map_from(groups.misc.Cactus(7))
+            False
         """
         if isinstance(G, SymmetricGroup):
             D = G.domain()
@@ -6825,6 +6830,9 @@ def _coerce_map_from_(self, G):
             return self._from_permutation_group_element
         if isinstance(G, StandardPermutations_n) and G.n <= self.n:
             return True
+        from sage.groups.cactus_group import CactusGroup
+        if isinstance(G, CactusGroup) and G.n() <= self.n:
+            return self._from_cactus_group_element
         return super()._coerce_map_from_(G)
 
     def _from_permutation_group_element(self, x):
@@ -6841,6 +6849,21 @@ def _from_permutation_group_element(self, x):
         """
         return self(x.domain())
 
+    def _from_cactus_group_element(self, x):
+        """
+        Return an element of ``self`` from a cactus group element.
+
+        EXAMPLES::
+
+            sage: J3 = groups.misc.Cactus(3)
+            sage: s12,s13,s23 = J3.gens()
+            sage: elt = s12 * s23 * s13
+            sage: P5 = Permutations(5)
+            sage: P5._from_cactus_group_element(elt)
+            [1, 3, 2, 4, 5]
+        """
+        return self(x.to_permutation())
+
     def as_permutation_group(self):
         """
         Return ``self`` as a permutation group.