Trac #33974: Documentation addition for symmetric functions - Cauchy identity

Include an example of how to use plethysm and tensor product of
symmetric functions in two sets of variables to verify the Cauchy
identities.

URL: https://trac.sagemath.org/33974
Reported by: tkarn
Ticket author(s): Trevor K. Karn
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/combinat/sf/sfa.py

@@ -3039,6 +3039,17 @@ def plethysm(self, x, include=None, exclude=None):
             sage: s[1,1,1](X*Y)
             s[1, 1, 1] # s[3] + s[2, 1] # s[2, 1] + s[3] # s[1, 1, 1]
 
+        One can use this to work with symmetric functions in two sets of
+        commuting variables. For example, we verify the Cauchy identities (in
+        degree 5)::
+
+            sage: m = SymmetricFunctions(QQ).m()
+            sage: P5 = Partitions(5)
+            sage: sum(s[mu](X)*s[mu](Y) for mu in P5) == sum(m[mu](X)*h[mu](Y) for mu in P5)
+            True
+            sage: sum(s[mu](X)*s[mu.conjugate()](Y) for mu in P5) == sum(m[mu](X)*e[mu](Y) for mu in P5)
+            True
+
         .. SEEALSO::
 
             :meth:`frobenius`