def seminormal_test(n):
"""
Runs a variety of tests to verify that the construction of the
seminormal basis works as desired. The numbers appearing are
Theorems in James and Kerber's 'Representation Theory of the
Symmetric Group'.
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import seminormal_test
sage: seminormal_test(3)
True
"""
for part in partition.Partitions_n(n):
for tab in StandardTableaux(part):
#3.1.10
if not e(tab) * (1 / kappa(part)) - e_hat(tab) == 0:
raise ValueError, "3.1.10 - %s" % tab
#3.2.12.2
value = e(tab) * epsilon(tab, 1) * e(tab) - e(tab) * (kappa(part))
if value != 0:
print value
raise ValueError, "3.2.12.2 - %s" % tab
for tab2 in StandardTableaux(part):
#3.2.8 1
if e_ik(tab, tab2) - e(tab) * pi_ik(
tab, tab2) * e(tab2) * (1 / kappa(part)) != 0:
raise ValueError, "3.2.8.1 - %s, %s" % (tab, tab2)
#3.2.8.1
if e(tab) * e_ik(tab, tab2) - e_ik(tab,
tab2) * (kappa(part)) != 0:
raise ValueError, "3.2.8.2 - %s, %s" % (tab, tab2)
if tab == tab2:
continue
if tab.last_letter_lequal(tab2):
#3.1.20
if e(tab2) * e(tab) != 0:
raise ValueError, "3.1.20 - %s, %s" % (tab, tab2)
if e_hat(tab2) * e_hat(tab) != 0:
raise ValueError, "3.1.20 - %s, %s" % (tab, tab2)
return True
def kappa(alpha):
r"""
Returns `\kappa_\alpha` which is n! divided by the number
of standard tableaux of shape `\alpha`.
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import kappa
sage: kappa(Partition([2,1]))
3
sage: kappa([2,1])
3
"""
try:
n = alpha.size()
except AttributeError:
n = sum(alpha)
return factorial(n)/StandardTableaux(alpha).cardinality()