Exploring symmetric function featuresΒΆ

[1]:
%display latex
[2]:
S = SymmetricFunctions(QQ)
[3]:
S.inject_shorthands()
/opt/sage-git2/local/lib/python2.7/site-packages/sage/combinat/sf/sf.py:1416: RuntimeWarning: redefining global value `e`
  inject_variable(shorthand, getattr(self, shorthand)())
[4]:
(s[2,1] + 3 * m[2,1]) * (e[3,1]+1)
[4]:
[5]:
print latex(s[4,3,2] * s[10,5,3])
s_{10,5,4,3,3,2} + s_{10,5,4,4,2,2} + s_{10,5,4,4,3,1} + s_{10,5,5,3,2,2} + s_{10,5,5,3,3,1} + s_{10,5,5,4,2,1} + s_{10,5,5,4,3} + s_{10,6,3,3,3,2} + 2s_{10,6,4,3,2,2} + 2s_{10,6,4,3,3,1} + 2s_{10,6,4,4,2,1} + s_{10,6,4,4,3} + s_{10,6,5,2,2,2} + 3s_{10,6,5,3,2,1} + 2s_{10,6,5,3,3} + s_{10,6,5,4,1,1} + 2s_{10,6,5,4,2} + s_{10,6,6,2,2,1} + s_{10,6,6,3,1,1} + 2s_{10,6,6,3,2} + s_{10,6,6,4,1} + s_{10,7,3,3,2,2} + s_{10,7,3,3,3,1} + s_{10,7,4,2,2,2} + 3s_{10,7,4,3,2,1} + 2s_{10,7,4,3,3} + s_{10,7,4,4,1,1} + 2s_{10,7,4,4,2} + 2s_{10,7,5,2,2,1} + 2s_{10,7,5,3,1,1} + 4s_{10,7,5,3,2} + 2s_{10,7,5,4,1} + s_{10,7,6,2,1,1} + 2s_{10,7,6,2,2} + 3s_{10,7,6,3,1} + s_{10,7,6,4} + s_{10,7,7,2,1} + s_{10,7,7,3} + s_{10,8,3,3,2,1} + s_{10,8,3,3,3} + s_{10,8,4,2,2,1} + s_{10,8,4,3,1,1} + 3s_{10,8,4,3,2} + s_{10,8,4,4,1} + s_{10,8,5,2,1,1} + 2s_{10,8,5,2,2} + 3s_{10,8,5,3,1} + s_{10,8,5,4} + 2s_{10,8,6,2,1} + 2s_{10,8,6,3} + s_{10,8,7,2} + s_{10,9,3,3,2} + s_{10,9,4,2,2} + s_{10,9,4,3,1} + s_{10,9,5,2,1} + s_{10,9,5,3} + s_{10,9,6,2} + s_{11,5,3,3,3,2} + 2s_{11,5,4,3,2,2} + 2s_{11,5,4,3,3,1} + 2s_{11,5,4,4,2,1} + s_{11,5,4,4,3} + s_{11,5,5,2,2,2} + 3s_{11,5,5,3,2,1} + 2s_{11,5,5,3,3} + s_{11,5,5,4,1,1} + 2s_{11,5,5,4,2} + 2s_{11,6,3,3,2,2} + 2s_{11,6,3,3,3,1} + 2s_{11,6,4,2,2,2} + 6s_{11,6,4,3,2,1} + 3s_{11,6,4,3,3} + 2s_{11,6,4,4,1,1} + 3s_{11,6,4,4,2} + 4s_{11,6,5,2,2,1} + 4s_{11,6,5,3,1,1} + 7s_{11,6,5,3,2} + 3s_{11,6,5,4,1} + 2s_{11,6,6,2,1,1} + 3s_{11,6,6,2,2} + 4s_{11,6,6,3,1} + s_{11,6,6,4} + s_{11,7,3,2,2,2} + 3s_{11,7,3,3,2,1} + 2s_{11,7,3,3,3} + 4s_{11,7,4,2,2,1} + 4s_{11,7,4,3,1,1} + 7s_{11,7,4,3,2} + 3s_{11,7,4,4,1} + 4s_{11,7,5,2,1,1} + 6s_{11,7,5,2,2} + 8s_{11,7,5,3,1} + 2s_{11,7,5,4} + s_{11,7,6,1,1,1} + 6s_{11,7,6,2,1} + 4s_{11,7,6,3} + s_{11,7,7,1,1} + 2s_{11,7,7,2} + s_{11,8,3,2,2,1} + s_{11,8,3,3,1,1} + 3s_{11,8,3,3,2} + 2s_{11,8,4,2,1,1} + 4s_{11,8,4,2,2} + 5s_{11,8,4,3,1} + s_{11,8,4,4} + s_{11,8,5,1,1,1} + 6s_{11,8,5,2,1} + 4s_{11,8,5,3} + 2s_{11,8,6,1,1} + 4s_{11,8,6,2} + s_{11,8,7,1} + s_{11,9,3,2,2} + s_{11,9,3,3,1} + 2s_{11,9,4,2,1} + s_{11,9,4,3} + s_{11,9,5,1,1} + 2s_{11,9,5,2} + s_{11,9,6,1} + s_{12,5,3,3,2,2} + s_{12,5,3,3,3,1} + s_{12,5,4,2,2,2} + 3s_{12,5,4,3,2,1} + 2s_{12,5,4,3,3} + s_{12,5,4,4,1,1} + 2s_{12,5,4,4,2} + 2s_{12,5,5,2,2,1} + 2s_{12,5,5,3,1,1} + 4s_{12,5,5,3,2} + 2s_{12,5,5,4,1} + s_{12,6,3,2,2,2} + 3s_{12,6,3,3,2,1} + 2s_{12,6,3,3,3} + 4s_{12,6,4,2,2,1} + 4s_{12,6,4,3,1,1} + 7s_{12,6,4,3,2} + 3s_{12,6,4,4,1} + 4s_{12,6,5,2,1,1} + 6s_{12,6,5,2,2} + 8s_{12,6,5,3,1} + 2s_{12,6,5,4} + s_{12,6,6,1,1,1} + 5s_{12,6,6,2,1} + 3s_{12,6,6,3} + 2s_{12,7,3,2,2,1} + 2s_{12,7,3,3,1,1} + 4s_{12,7,3,3,2} + 4s_{12,7,4,2,1,1} + 6s_{12,7,4,2,2} + 8s_{12,7,4,3,1} + 2s_{12,7,4,4} + 2s_{12,7,5,1,1,1} + 10s_{12,7,5,2,1} + 6s_{12,7,5,3} + 4s_{12,7,6,1,1} + 6s_{12,7,6,2} + 2s_{12,7,7,1} + s_{12,8,3,2,1,1} + 2s_{12,8,3,2,2} + 3s_{12,8,3,3,1} + s_{12,8,4,1,1,1} + 6s_{12,8,4,2,1} + 4s_{12,8,4,3} + 4s_{12,8,5,1,1} + 6s_{12,8,5,2} + 4s_{12,8,6,1} + s_{12,8,7} + s_{12,9,3,2,1} + s_{12,9,3,3} + s_{12,9,4,1,1} + 2s_{12,9,4,2} + 2s_{12,9,5,1} + s_{12,9,6} + s_{13,5,3,3,2,1} + s_{13,5,3,3,3} + s_{13,5,4,2,2,1} + s_{13,5,4,3,1,1} + 3s_{13,5,4,3,2} + s_{13,5,4,4,1} + s_{13,5,5,2,1,1} + 2s_{13,5,5,2,2} + 3s_{13,5,5,3,1} + s_{13,5,5,4} + s_{13,6,3,2,2,1} + s_{13,6,3,3,1,1} + 3s_{13,6,3,3,2} + 2s_{13,6,4,2,1,1} + 4s_{13,6,4,2,2} + 5s_{13,6,4,3,1} + s_{13,6,4,4} + s_{13,6,5,1,1,1} + 6s_{13,6,5,2,1} + 4s_{13,6,5,3} + 2s_{13,6,6,1,1} + 3s_{13,6,6,2} + s_{13,7,3,2,1,1} + 2s_{13,7,3,2,2} + 3s_{13,7,3,3,1} + s_{13,7,4,1,1,1} + 6s_{13,7,4,2,1} + 4s_{13,7,4,3} + 4s_{13,7,5,1,1} + 6s_{13,7,5,2} + 4s_{13,7,6,1} + s_{13,7,7} + 2s_{13,8,3,2,1} + 2s_{13,8,3,3} + 2s_{13,8,4,1,1} + 4s_{13,8,4,2} + 4s_{13,8,5,1} + 2s_{13,8,6} + s_{13,9,3,2} + s_{13,9,4,1} + s_{13,9,5} + s_{14,5,3,3,2} + s_{14,5,4,2,2} + s_{14,5,4,3,1} + s_{14,5,5,2,1} + s_{14,5,5,3} + s_{14,6,3,2,2} + s_{14,6,3,3,1} + 2s_{14,6,4,2,1} + s_{14,6,4,3} + s_{14,6,5,1,1} + 2s_{14,6,5,2} + s_{14,6,6,1} + s_{14,7,3,2,1} + s_{14,7,3,3} + s_{14,7,4,1,1} + 2s_{14,7,4,2} + 2s_{14,7,5,1} + s_{14,7,6} + s_{14,8,3,2} + s_{14,8,4,1} + s_{14,8,5}
[6]:
s[3,2].coproduct()
[6]:
[7]:
tensor([s[3,2], (p[2,1]+p[3])])
[7]:
[8]:
p[3](s[2,1])
[8]: