A121741 Dimensions of the irreducible representations of the simple Lie algebra of type A2 (equivalently, the group SL3) over the complex numbers, listed in increasing order.

1, 3, 6, 8, 10, 15, 21, 24, 27, 28, 35, 36, 42, 45, 48, 55, 60, 63, 64, 66, 78, 80, 81, 90, 91, 99, 105, 120, 125, 132, 136, 143, 153, 154, 162, 165, 168, 171, 190, 192, 195, 210, 216, 224, 231, 234, 253, 255, 260, 270, 273, 276, 280, 288, 300

Offset: 1

Author

Skip Garibaldi (skip(AT)member.ams.org), Aug 19 2006, Aug 23 2006

Keywords

• nonn

Comments

We include "1" for the 1-dimensional trivial representation and we list each dimension once, ignoring the fact that inequivalent representations may have the same dimension.

Numbers of the form (x * (x - y) * (x - z) + y * (y - x) * (y - z) + z * (z - x) * (z - y)) / 18 with x + y + z = 0 and x * y * z > 0. - Michael Somos, Jun 26 2013

Positive numbers of the form (r-s)*r*(r+s) where r and s are integers, i.e., the product of three integers in arithmetic progression. In the expression above, set x = r-s, y = r+s, and z = -x-y. - Elliott Line, Dec 22 2020

References

• N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Springer, 2002.
• J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1997.

Links

• Andy Huchala, Table of n, a(n) for n = 1..20000
• Andy Huchala, Java Program
• Wikipedia, The special linear group

Crossrefs

Equals A088915(n+1)/2.

Cf. A121732, A121736, A121737, A121738, A121739, A104599, A121214, A162651.

Programs

GAP

# see program at sequence A121732

Python

from itertools import count, islice
from sympy import divisors, integer_nthroot
def A121741_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        for k in divisors(m<<1,generator=True):
            p, q = integer_nthroot(k**4+(k*m<<3),2)
            if q and not (p-k**2)%(k<<1):
                yield m
                break
A121741_list = list(islice(A121741_gen(),20)) # Chai Wah Wu, Jul 03 2023