| Displaying 1-10 of 10 results found.
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1, 2, 5, 10, 19, 31, 54, 77, 108, 155, 206, 274, 356, 458, 579, 715, 884, 1068, 1270, 1531, 1805
EXAMPLE
For n=3, we list the index a(3)=5 because A121736(5)=1463 equals A030649(3).
Dimensions of the irreducible representations of the simple Lie algebra of type E8 over the complex numbers, listed in increasing order.
+10
16
1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860
COMMENTS
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.
REFERENCES
J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1997.
FORMULA
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.
EXAMPLE
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.
PROG
(GAP) # see program given in link.
AUTHOR
Skip Garibaldi (skip(AT)mathcs.emory.edu), Aug 18 2006
Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in increasing order.
+10
10
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
COMMENTS
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.
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.
FORMULA
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.
EXAMPLE
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.
PROG
(GAP) # see program at sequence A121732
AUTHOR
Skip Garibaldi (skip(AT)member.ams.org), Aug 19 2006
Dimensions of the irreducible representations of the simple Lie algebra of type G2 over the complex numbers, listed in increasing order.
+10
9
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
COMMENTS
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.
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.
FORMULA
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.
EXAMPLE
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.
PROG
(GAP) # see program at sequence A121732
AUTHOR
Skip Garibaldi (skip(AT)member.ams.org), Aug 19 2006
Dimensions of the irreducible representations of the simple Lie algebra of type F4 over the complex numbers, listed in increasing order.
+10
9
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
COMMENTS
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.
REFERENCES
N. Bourbaki, Lie groups and Lie algebras, Chapter 4-6, Springer, 2002.
J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1997.
FORMULA
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.
EXAMPLE
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.
AUTHOR
Skip Garibaldi (skip(AT)member.ams.org), Aug 19 2006
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.
+10
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
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.
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
AUTHOR
Skip Garibaldi (skip(AT)member.ams.org), Aug 19 2006, Aug 23 2006
Dimensions of the irreducible representations of the simple Lie algebra of type D4 over the complex numbers, listed in increasing order.
+10
8
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
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.
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.
FORMULA
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.
EXAMPLE
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.
PROG
(GAP) # see program at sequence A121732
AUTHOR
Skip Garibaldi (skip(AT)member.ams.org), Aug 19 2006
Dimensions of multiples of minimal representation of complex Lie algebra E7.
+10
3
1, 56, 1463, 24320, 293930, 2785552, 21737254, 144538624, 839848450, 4347450800, 20355385710, 87265194240, 345992859975, 1279301331000, 4442249264625, 14573017267200, 45398364338250, 134897996890800, 383822534859750, 1049290591104000, 2764459117589400
COMMENTS
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)
REFERENCES
Onishchik and Vinberg, Seminar on Lie Groups and Algebraic Groups, Springer Verlag 1990, see Table 5.
FORMULA
a(n) = (1/10950439500)*(n+9)*binomial(n+17, 4)*binomial(n+4, 4)*binomial(n+13, 9)^2.
MAPLE
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)];
MATHEMATICA
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 *)
PROG
(PARI) 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
AUTHOR
Paolo Dominici (pl.dm(AT)libero.it)
List of dimensions for which there exist several non-isomorphic irreducible representations of E7.
+10
1
1903725824, 16349520330, 8971740610560, 34695403142400, 824608512000000, 4660749155462400, 5099341625414400, 6681177699123200, 35516286743137200, 61732518862014000, 95583619816439040, 631645584845184000, 972524604841574400, 1199167756428096000
COMMENTS
Terms which could be repeated in A121736. There are infinitely many terms in this sequence; see A181746.
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.
EXAMPLE
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.
PROG
(Java) // See Links section of A181746. (C++) // See Links section above and in A181746.
Dimensions of the simple Lie algebras over complex numbers (with repetitions), sorted nondecreasingly.
+10
0
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
COMMENTS
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.
REFERENCES
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.
LINKS
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.
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