A237362 Nonzero Hilbert function values for the invariant ring of 3 X 3 X 3 X 3 tensors.

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.

Links

Table of n, a(n) for n=1..15.

Hanspeter Kraft & Claudio Procesi, Classical invariant theory, a primer

Luke Oeding, Maple program for A237362 (provided by Murray Bremner)

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

Adjacent sequences: A237359 A237360 A237361 * A237363 A237364 A237365

Keyword

nonn,hard,more

Author

Luke Oeding, Feb 06 2014

Status

approved