A121741 - OEIS

%I #36 Jul 04 2023 03:55:52

%S 1,3,6,8,10,15,21,24,27,28,35,36,42,45,48,55,60,63,64,66,78,80,81,90,

%T 91,99,105,120,125,132,136,143,153,154,162,165,168,171,190,192,195,

%U 210,216,224,231,234,253,255,260,270,273,276,280,288,300

%N 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.

%C 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.

%C 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

%C 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

%D N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Springer, 2002.

%D J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1997.

%H Andy Huchala, <a href="/A121741/b121741.txt">Table of n, a(n) for n = 1..20000</a>

%H Andy Huchala, <a href="/A121741/a121741.java.txt">Java Program</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Special_linear_group">The special linear group</a>

%o (GAP) # see program at sequence A121732

%o (Python)

%o from itertools import count, islice

%o from sympy import divisors, integer_nthroot

%o def A121741_gen(startvalue=1): # generator of terms >= startvalue

%o     for m in count(max(startvalue,1)):

%o         for k in divisors(m<<1,generator=True):

%o             p, q = integer_nthroot(k**4+(k*m<<3),2)

%o             if q and not (p-k**2)%(k<<1):

%o                 yield m

%o                 break

%o A121741_list = list(islice(A121741_gen(),20)) # _Chai Wah Wu_, Jul 03 2023

%Y Equals A088915(n+1)/2.

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

%K nonn

%O 1,2

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