mjo/clan/normal_decomposition_element.py: more Ishi IP tests

Test the value of the beta_{k} when we have normalized the
off-diagonal basis elements using the Ishi norm.

diff --git a/mjo/clan/normal_decomposition_element.py b/mjo/clan/normal_decomposition_element.py
index 2c4140faca3eca557f569cffa97a3f903f60b150..59d0b9cc35d3330f106f53237b35739f2c264b1d 100644 (file)
--- a/mjo/clan/normal_decomposition_element.py
+++ b/mjo/clan/normal_decomposition_element.py
@@ -46,6 +46,20 @@ class NormalDecompositionElement(ClanElement):
         ....:      if (y := C.random_element().elt(i, r)) )
         True
 
+    Compute the `beta_{k}` scalars from the same section::
+
+        sage: C = random_clan(nontrivial=True)
+        sage: while C.rank() < 2:
+        ....:     # need rank >= 2 for the off-diagonal to exist
+        ....:     C = random_clan(nontrivial=True)
+        sage: r = 0  # Ishi convention
+        sage: all( (u*u).terms() == [2*C.idempotent(i)]
+        ....:      for i in range(1, C.rank())
+        ....:      for u in (b/b.norm()
+        ....:                for b in C.basis()
+        ....:                if b.elt(i, r) == b) )
+        True
+
     """
 
     def tr(self):