login
A237362
Nonzero Hilbert function values for the invariant ring of 3 X 3 X 3 X 3 tensors.
1
1, 4, 23, 136, 1147, 11277, 105046, 927054, 7581063, 57507712, 405267267, 2662585198, 16366889904, 94486124304, 514135539077
OFFSET
1,2
COMMENTS
The values of the Hilbert function H(d) for the invariant ring for 3 X 3 X 3 X 3 tensors were computed using standard character theory.
We know that H(3n+1) = H(3n+2) = 0 and the sequence we are interested in is defined by a(n) := H(3n).
REFERENCES
J.-L. Brylinski & R. Brylinski, Invariant polynomial functions on k qudits. Mathematics of Quantum Computation, pages 277-286. Chapman & Hall/CRC, Boca Raton, FL, 2002.
William Fulton & Joe Harris, Representation Theory: A First Course, Springer Verlag, 1991.
H. Derksen, G. Kemper, Computational Invariant Theory, Encyclopaedia of Mathematical Sciences, R. V. Gamkrelidze, V. L. Popov subseries eds., Invariant Theory and Algebraic Transformation Groups I, Springer Verlag, Berlin, Heidelberg, New York, 2002.
FORMULA
a(n) = H(3n); H(d) = (1/d!) sum_{lambda} C_lambda * chi(mu,lambda)^k.
The sum is over all partitions lambda of d, and C_lambda is the conjugacy class of cycle type lambda, and chi(mu,lambda) is the value of the character of the representation mu=[n,n,n] on C_lambda and k=4. This formula is found, for instance in [Brylinski-Brylinski](Prop. 2.1)
MAPLE
# see link above
CROSSREFS
Sequence in context: A038723 A302761 A091640 * A067110 A290052 A158197
KEYWORD
nonn,hard,more
AUTHOR
Luke Oeding, Feb 06 2014
STATUS
approved