A121741 -id:A121741 - cp's OEIS frontend

A121732 Dimensions of the irreducible representations of the simple Lie algebra of type E8 over the complex numbers, listed in increasing order.

1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860

Offset: 1

Skip Garibaldi (skip(AT)mathcs.emory.edu), Aug 18 2006

nonn

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

Inequivalent representations can have the same dimension. For example, the highest weights 10100000 and 10000011 (with fundamental weights numbered as in Bourbaki) both correspond to irreducible representations of dimension 8634368000.

			The highest weight 00000000 corresponds to the 1-dimensional module on which E8 acts trivially. The smallest faithful representation of E8 is the adjoint representation of dimension 248 (the second term in the sequence), with highest weight 00000001. The smallest non-fundamental representation has dimension 27000 (the fourth term), corresponding to the highest weight 00000002.

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

Andy Huchala, Table of n, a(n) for n = 1..20000
Skip Garibaldi, Gap program
Wikipedia, E8 (mathematics)

Cf. A121736, A121737, A121738, A121739, A104599, A121741, A121214, A030650.

GAP
```
# see program given in link.
```

Given a vector of 8 nonnegative integers, the Weyl dimension formula tells you the dimension of the corresponding irreducible representation. The list of such dimensions is then sorted numerically.

A121736 Dimensions of the irreducible representations of the simple Lie algebra of type E7 over the complex numbers, listed in increasing order.

Original entry on oeis.org

1, 56, 133, 912, 1463, 1539, 6480, 7371, 8645, 24320, 27664, 40755, 51072, 86184, 150822, 152152, 238602, 253935, 293930, 320112, 362880, 365750, 573440, 617253, 861840, 885248, 915705, 980343, 2273920, 2282280, 2785552, 3424256, 3635840

Offset: 1

Skip Garibaldi (skip(AT)member.ams.org), Aug 18 2006

nonn

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

A121737 Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in increasing order.

Original entry on oeis.org

1, 27, 78, 351, 650, 1728, 2430, 2925, 3003, 5824, 7371, 7722, 17550, 19305, 34398, 34749, 43758, 46332, 51975, 54054, 61425, 70070, 78975, 85293, 100386, 105600, 112320, 146432, 252252, 314496, 359424, 371800, 386100, 393822, 412776, 442442

Offset: 1

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

nonn

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

			The highest weight 000000 corresponds to the 1-dimensional module on which E6 acts trivially. The smallest faithful representations of E6 have dimension 27, highest weight 000001 or 100000 and are minuscule. The adjoint representation of dimension 78 (the third term in the sequence) has highest weight 010000.

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

Andy Huchala, Table of n, a(n) for n = 1..20000
Andy Huchala, Java program
Sudeep Podder and Parameswaran Sankaran, The K-ring of E_6/Spin(10), arXiv:2307.04844 [math.KT], 2023.
Wikipedia, E_6 (mathematics)

Cf. A121732, A121736, A121738, A121739, A104599, A121741.

GAP
```
# see program at sequence A121732
```

Given a vector of 6 nonnegative integers, the Weyl dimension formula tells you the dimension of the corresponding irreducible representation. The list of such dimensions is then sorted numerically.

A121738 Dimensions of the irreducible representations of the simple Lie algebra of type F4 over the complex numbers, listed in increasing order.

Original entry on oeis.org

1, 26, 52, 273, 324, 1053, 1274, 2652, 4096, 8424, 10829, 12376, 16302, 17901, 19278, 19448, 29172, 34749, 76076, 81081, 100776, 106496, 107406, 119119, 160056, 184756, 205751, 212992, 226746, 340119, 342056, 379848, 412776, 420147, 627912

Offset: 1

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

nonn

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

			The highest weight 0000 corresponds to the 1-dimensional module on which F4 acts trivially. The smallest faithful representation of F4 is the "standard" representation of dimension 26 (the second term in the sequence), with highest weight 0001. (This representation is typically viewed as the trace zero elements in a 27-dimensional exceptional Jordan algebra.) The adjoint representation has dimension 52 (the third term in the sequence) and highest weight 1000.

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

Andy Huchala, Table of n, a(n) for n = 1..20000
Andy Huchala, Java program
Wikipedia, F_4 (mathematics)

Cf. A121732, A121736, A121737, A121739, A104599, A121741.

GAP
```
# see program at A121732
```

Given a vector of 4 nonnegative integers, the Weyl dimension formula tells you the dimension of the corresponding irreducible representation. The list of such dimensions is then sorted numerically.

A088915 Nonnegative numbers of the form mn(m+n) with integers m,n.

Original entry on oeis.org

0, 2, 6, 12, 16, 20, 30, 42, 48, 54, 56, 70, 72, 84, 90, 96, 110, 120, 126, 128, 132, 156, 160, 162, 180, 182, 198, 210, 240, 250, 264, 272, 286, 306, 308, 324, 330, 336, 342, 380, 384, 390, 420, 432, 448, 462, 468, 506, 510, 520, 540, 546, 552, 560, 576, 600

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 29 2003

nonn

These are the values of 3 X 3 Vandermonde determinants with integer entries.
Solutions (m,n) are integral points on the elliptic curve m*n*(m+n)=a(n). Entries with record number of solutions are: 2, 6, 30, 240, 6480, 18480, 147840, 3991680 Possibly not minimal: a(n)=988159766157083520000000 has 22 solutions a(n)=2880932262848640000 20 solutions Multiplication of a(n) by u^3 does not decrease the number of solutions. [From Georgi Guninski, Oct 25 2010]
Contribution from R. J. Mathar, Oct 24 2010: (Start)
Examples of entries with more than one representation are:
- 30 = 5*1*6 = 3*2*5,
- 240 = 15*1*16 = 10*2*12 = 6*4*10, 6480 = 80*1*81 = 45*3*48 = 30*6*36 = 18*12*30,
- 18408 = 77*3*80 = 66*4*70 = 48*7*55 = 30*14*44 = 22*20*42,
- 147840 = 384*1*385 = 154*6*160 = 132*8*140 = 96*14*110 = 60*28*88 = 44*40*84 (6 representations),
- 110270160 = 6*4284*4290 = 60*1326*1386 = 66*1260*1326 = 102*990*1092 = ... with 8 representations. (End)

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A121741.

filter:= proc(n) local d,nd,x,y;
  d:= numtheory:-divisors(n);
  nd:= nops(d);
  for x from 1 to nd do
    for y from 1 to x do
      if d[x]*d[y]*(d[x]+d[y])=n then return true fi
  od od:
  false
end proc:
filter(0):= 0:
select(filter, [seq(i,i=0..1000,2)]); # Robert Israel, Jul 24 2018

Select[Range[0, 600], {} != FindInstance[m n (m + n) == # && n >= 0 && m >= 0, {m, n}, Integers, 1] &] (* Giovanni Resta, Jul 24 2018 *)

from itertools import count, islice
from sympy import divisors, integer_nthroot
def A088915_gen(startvalue=0): # generator of terms >= startvalue
    for m in count(max(startvalue,0)):
        if m == 0:
            yield m
        else:
            for k in divisors(m,generator=True):
                p, q = integer_nthroot(k**4+(k*m<<2),2)
                if q and not (p-k**2)%(k<<1):
                    yield m
                    break
A088915_list = list(islice(A088915_gen(),20)) # Chai Wah Wu, Jul 03 2023

a(n) = 2 * A121741(n-1) for n>=2.

More terms from Hugo Pfoertner, Apr 10 2004

A121214 Dimensions of the irreducible representations of the algebraic group SL4 (equivalently, simple Lie algebra of type A3) over the complex numbers, listed in increasing order.

Original entry on oeis.org

1, 4, 6, 10, 15, 20, 35, 36, 45, 50, 56, 60, 64, 70, 84, 105, 120, 126, 140, 160, 165, 175, 189, 196, 216, 220, 224, 256, 270, 280, 286, 300, 315, 336, 360, 364, 384, 396, 420, 440, 455, 480, 500, 504, 540, 560, 594, 616, 630, 640, 680, 715, 729, 735, 750, 756

Offset: 1

Skip Garibaldi (skip(AT)member.ams.org), Aug 20 2006

nonn

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.

			The highest weight 000 corresponds to the 1-dimensional module on which SL4 acts trivially. The standard representation and its dual have dimension 4 (the second term in the sequence) and highest weights 100 and 001. The third term in the sequence, 6, is the dimension of the representation of SL4 on the second exterior power of the standard representation; it has highest weight 010. The fourth term, 10, is the dimension of the second symmetric power of the standard representation or its dual, with highest weight 200 or 002. The fifth term, 15, corresponds to the adjoint representation with highest weight 101.

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

Andy Huchala, Table of n, a(n) for n = 1..20000
Andy Huchala, Java program
Wikipedia, Special linear group

Cf. A121732, A121741.

GAP
```
# see program at A121732
```

Given a vector of 3 nonnegative integers, the Weyl dimension formula tells you the dimension of the corresponding irreducible representation. The list of such dimensions is then sorted numerically.

A263005 Dimensions of the simple Lie algebras over complex numbers (with repetitions), sorted nondecreasingly.

Original entry on oeis.org

3, 8, 10, 14, 15, 21, 21, 24, 28, 35, 36, 36, 45, 48, 55, 55, 57, 63, 66, 78, 78, 78, 80, 91, 99, 105, 105, 120, 120, 133, 136, 136, 143, 153, 168, 171, 171, 190, 195, 210, 210, 224, 231, 248, 253, 253, 255, 276, 288, 300, 300

Offset: 1

Wolfdieter Lang, Oct 23 2015

This sequence gives the dimensions of the (compact) simple Lie algebras A_l, l >= 1, B_l, l >= 2, C_l >= 3, D_l, l >= 4, E_6, E_7, E_8, F_4 and G_2 which are l*(l+2), l*(2*l + 1), l*(2*l + 1), l*(2*l - 1), 78, 133, 248, 52 and 14, respectively. These are also the dimensions of the adjoint representations of these Lie algebras. For the l-ranges see the Humphreys reference, p. 58, and for the dimensions, e.g., the Samelson link, Theorem A, p. 74.

The dimension duplications occur for the B_l and C_l series for l >= 3.

E. Cartan, Sur la structure des groupes de transformation finis et continus. Thèse Paris 1894. Oeuvres Complètes, I,1, pp. 137-287, Paris 1952.
J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1972.

W. Killing, Die Zusammensetzung der stetigen endlichen Transformationsgruppen, Mathematische Ann. I: 31 (1888) 252-290, II: 33 (1889) 1-48, III: 34 (1889) 57-122, IV: 36 (1890) 161-189: I, II, III, IV.
Hans Samelson, Notes on Lie Algebras.

Cf. A104599, A121214, A121732, A121736, A121737, A121738, A121739, A121741.

A333821 Numbers k that can be represented in the form k = p^3 - q^3 - r^3, where p, q, r are positive integers satisfying p = q + r.

Original entry on oeis.org

6, 18, 36, 48, 60, 90, 126, 144, 162, 168, 210, 216, 252, 270, 288, 330, 360, 378, 384, 396, 468, 480, 486, 540, 546, 594, 630, 720, 750, 792, 816, 858, 918, 924, 972, 990, 1008, 1026, 1140, 1152, 1170, 1260, 1296, 1344, 1386, 1404, 1518, 1530, 1560, 1620, 1638, 1656, 1680, 1728, 1800

Offset: 1

Antonio Roldán, Apr 06 2020

nonn

An alternative representation of k is k = 3*q*r*(q+r), with q, r positive integers, then k is a multiple of 6.

			60 is in the sequence because 60 = 5^3 - 4^3 - 1^3, with 5 = 4 + 1.

Cf. A000578, A003072, A024975, A078129, A255018, A275154.

ok(n) = {my(i=1, a=0, m=0, j); if(n%6==0, while(a<=n&&m==0, j=1; while(j

a(n) = 6 * A121741(n).

cp's OEIS Frontend

A121732 Dimensions of the irreducible representations of the simple Lie algebra of type E8 over the complex numbers, listed in increasing order.

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A121736 Dimensions of the irreducible representations of the simple Lie algebra of type E7 over the complex numbers, listed in increasing order.

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A121737 Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in increasing order.

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A121738 Dimensions of the irreducible representations of the simple Lie algebra of type F4 over the complex numbers, listed in increasing order.

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GAP

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A088915 Nonnegative numbers of the form mn(m+n) with integers m,n.

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Maple

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A121214 Dimensions of the irreducible representations of the algebraic group SL4 (equivalently, simple Lie algebra of type A3) over the complex numbers, listed in increasing order.

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Examples

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Links

Crossrefs

Programs

GAP

Formula

A263005 Dimensions of the simple Lie algebras over complex numbers (with repetitions), sorted nondecreasingly.

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