A353173 Dimension of space of invariants of n-th tensor power of the 26-dimensional fundamental (or "standard") irreducible representation of F_4.

1, 0, 1, 1, 5, 15, 70, 330, 1820, 10858, 70875, 497135, 3727955, 29658410, 248989676, 2194891440, 20231692430, 194286848280, 1937546532820, 20008993160460, 213436182918652, 2346406693816315, 26531060178217182, 307987244037724262, 3664579007885995952

Offset

0,5

Comments

It is known that a(n) satisfies a linear recurrence relation with polynomial coefficients. The limit of a(n+1)/a(n) is 26.

Links

Table of n, a(n) for n=0..24.

MathOverflow, Invariants for the exceptional complex simple Lie algebra F4

Example

a(1)=0 because there is no F_4-invariant linear form on the 26-dimensional representation; a(2)=1 because there is, up to scalars, precisely one invariant quadratic form.

Prog

(LiE) p_tensor(n, [0, 0, 0, 1], F4)|[0, 0, 0, 0] # Returns the value of a(n).

Crossrefs

The analogous sequence for the (52-dimensional) adjoint representation of F_4 is: A179685.

A similar sequence for G_2 (for its 7-dimensional fundamental irreducible representation) is: A059710.

A similar sequence for B_2 (for its standard 5-dimensional irreducible representation) is: A095922.

For A_n the similar sequence (omitting some 0's) is given by the (n+1)-dimensional Catalan numbers, e.g., A005789 for A_2.

Sequence in context: A149638 A149639 A149640 * A355603 A355507 A259205

Adjacent sequences: A353170 A353171 A353172 * A353174 A353175 A353176

Keyword

nonn

Author

David A. Madore, Apr 28 2022

Status

approved