A121736 -id:A121736 - cp's OEIS frontend

A132210 a(n) = index j such that A121736(j)=A030649(n).

1, 2, 5, 10, 19, 31, 54, 77, 108, 155, 206, 274, 356, 458, 579, 715, 884, 1068, 1270, 1531, 1805

Offset: 1

Alexander R. Povolotsky, Nov 06 2007

nonn

A030649 is a subset of A121736 (A. R. Povolotsky, Nov 06 2007). - R. J. Mathar, Nov 16 2007

			For n=3, we list the index a(3)=5 because A121736(5)=1463 equals A030649(3).

Cf. A030649, A121736.

A121736(a(n))=A030649(n). - R. J. Mathar, Nov 16 2007

Better definition from R. J. Mathar, Nov 16 2007

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

Original entry on oeis.org

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.

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.

A104599 Dimensions of the irreducible representations of the simple Lie algebra of type G2 over the complex numbers, listed in increasing order.

Original entry on oeis.org

1, 7, 14, 27, 64, 77, 182, 189, 273, 286, 378, 448, 714, 729, 748, 896, 924, 1254, 1547, 1728, 1729, 2079, 2261, 2926, 3003, 3289, 3542, 4096, 4914, 4928, 5005, 5103, 6630, 7293, 7371, 7722, 8372, 9177, 9660, 10206, 10556, 11571, 11648, 12096, 13090

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 00 corresponds to the 1-dimensional module on which G2 acts trivially. The smallest faithful representation of G2 is the "standard" representation of dimension 7 (the second term in the sequence), with highest weight 10. (This vector space can be viewed as the trace zero elements of an octonion algebra.) The third term in the sequence, 14, is the dimension of the adjoint representation, which has highest weight 01.

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, G_2 (mathematics)

Cf. A121732, A121736, A121737, A121738, A121739.

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

Given a vector of 2 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.

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.

Original entry on oeis.org

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

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

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, The special linear group

Equals A088915(n+1)/2.

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

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

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

A121739 Dimensions of the irreducible representations of the simple Lie algebra of type D4 over the complex numbers, listed in increasing order.

Original entry on oeis.org

1, 8, 28, 35, 56, 112, 160, 224, 294, 300, 350, 567, 672, 840, 1296, 1386, 1400, 1568, 1680, 1925, 2400, 2640, 2800, 3675, 3696, 4096, 4312, 4536, 4719, 5775, 6160, 6600, 7392, 7776, 7840, 8008, 8800, 8910, 8918, 10752, 12320, 12936, 13013, 13728, 15015

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 fact that inequivalent representations may have the same dimension.

			The highest weight 0000 corresponds to the 1-dimensional module on which D4 acts trivially. The second second term in the sequence is 8, corresponding to the three inequivalent representations with highest weights 1000, 0010 and 0001 respectively. The third term in the sequence is 28, corresponding to the adjoint representation, which has highest weight 0100.

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, Triality

Cf. A121732, A121736, A121737, A121738, A104599.

GAP
```
# see program at sequence 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.

A030649 Dimensions of multiples of minimal representation of complex Lie algebra E7.

Original entry on oeis.org

1, 56, 1463, 24320, 293930, 2785552, 21737254, 144538624, 839848450, 4347450800, 20355385710, 87265194240, 345992859975, 1279301331000, 4442249264625, 14573017267200, 45398364338250, 134897996890800, 383822534859750, 1049290591104000, 2764459117589400

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it)

nonn

From Alexander R. Povolotsky, Nov 19 2007: (Start)
After adjustment for the fact that a(n) is indexed from 0 while A121736 is indexed from 1, it appears that in many cases (with some exceptions) (a(n) - A121736(n+1))/133 (where A121736(3) = 133) yields integral values:
(1 - 1)/133 = 0
(56 - 56)/133 = 0
(1463 - 133) / 133 = 10
(24320 - 912) / 133 = 176
(293930 - 1463) / 133 = 2199
(2785552 - 1539) / 133 = 146527/7
(21737254 - 6480) / 133 = 21730774/133
(144538624 - 7371) / 133 = 144531253/133
(839848450 - 8645) / 133 = 6314585
(4347450800 - 24320) / 133 = 228811920/7
(20355385710 - 27664) / 133 = 153047805
(87265194240 - 40755) / 133 = 656128974
(345992859975 - 51072) / 133 = 2601449691
(1279301331000 - 86184) / 133 = 9618806352
(4442249264625 - 150822) / 133 = 233802584937/7
(14573017267200 - 152152)/133 = 109571557256
(45398364338250 - 238602)/133 = 341341083456
(134897996890800 - 253935)/133 = 1014270651405
(383822534565820 - 293930)/133 = 2885883718540
(1049290591104000 - 320112)/133 = 1049290590783888/133
...
Note that 133 is also the dimension of the Lie algebra E_7. (End)

Onishchik and Vinberg, Seminar on Lie Groups and Algebraic Groups, Springer Verlag 1990, see Table 5.

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.2(ii), case a=4]

Cf. A121736.

b:=binomial; t72b:= proc(a,k) ((a+k+1)/(a+1)) * b(k+2*a+1,k)*b(k+3*a/2+1,k)/(b(k+a/2,k)); end; [seq(t72b(8,k),k=0..28)];

Table[(1/10950439500)*(n + 9)*Binomial[n + 17, 4]*Binomial[n + 4, 4]* Binomial[n + 13, 9]^2, {n,0,50}] (* G. C. Greubel, Feb 19 2017 *)

for(n=0,25, print1((1/10950439500)*(n+9)*binomial(n+17, 4)*binomial(n+4, 4)*binomial(n+13, 9)^2, ", ")) \\ G. C. Greubel, Feb 19 2017

a(n) = (1/10950439500)*(n+9)*binomial(n+17, 4)*binomial(n+4, 4)*binomial(n+13, 9)^2.

Entry revised by N. J. A. Sloane, Oct 20 2007

A339251 List of dimensions for which there exist several non-isomorphic irreducible representations of E7.

Original entry on oeis.org

1903725824, 16349520330, 8971740610560, 34695403142400, 824608512000000, 4660749155462400, 5099341625414400, 6681177699123200, 35516286743137200, 61732518862014000, 95583619816439040, 631645584845184000, 972524604841574400, 1199167756428096000

Offset: 1

Andy Huchala, Apr 02 2021

nonn

Terms which could be repeated in A121736.

There are infinitely many terms in this sequence; see A181746.

			With the fundamental weights numbered as in Bourbaki, the irreducible E7-modules with highest weights [0,0,0,1,1,0,0] and [0,0,0,0,0,2,3] both have dimension 1903725824. The highest weights [3,0,0,1,0,0,0] and [0,0,0,0,1,0,5] both correspond to irreducible representations of dimension 16349520330.

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..9000
Andy Huchala, C++ Program
Wikipedia, E7 (mathematics)

Cf. A181746, A121736.

// See Links section of A181746.
(C++) // See Links section above and in A181746.

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.

cp's OEIS Frontend

A132210 a(n) = index j such that A121736(j)=A030649(n).

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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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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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A104599 Dimensions of the irreducible representations of the simple Lie algebra of type G2 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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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.

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Python

A121739 Dimensions of the irreducible representations of the simple Lie algebra of type D4 over the complex numbers, listed in increasing order.

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GAP