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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.
9
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
1,2
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
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Skip Garibaldi (skip(AT)member.ams.org), Aug 19 2006, Aug 23 2006
STATUS
approved