cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

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

Views

Author

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

Keywords

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.
See also comments in A030649.

Examples

			The highest weight 0000000 corresponds to the 1-dimensional module on which E7 acts trivially. The smallest faithful representation of E7 is the so-called "standard" representation of dimension 56 (the second term in the sequence), with highest weight 0000001; it is minuscule and supports the famous invariant quartic form. The adjoint representation of dimension 133 (the third term in the sequence), has highest weight 1000000.

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

Crossrefs

Cf. A121732, A121737, A121738, A121739, A104599, A121741, A030649.

Programs

  • GAP
    # see program given in sequence A121732

Formula

Given a vector of 7 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

Views

Author

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

Keywords

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.

Examples

			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.

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

Crossrefs

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

Programs

  • GAP
    # see program at sequence A121732

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.

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

Views

Author

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

Keywords

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.

Examples

			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.

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

Crossrefs

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

Programs

  • GAP
    # see program at sequence A121732

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.

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

Views

Author

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

Keywords

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.

Examples

			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.

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.

Links

Crossrefs

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

Programs

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.

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

Views

Author

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

Keywords

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

Crossrefs

Equals A088915(n+1)/2.
Cf. A121732, A121736, A121737, A121738, A121739, A104599, A121214, A162651.

Programs

  • 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

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

Views

Author

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

Keywords

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.

Examples

			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.

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

Crossrefs

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

Programs

  • GAP
    # see program at sequence A121732

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.

A181746 List of dimensions for which there exist several non-isomorphic irreducible representations of E8.

Original entry on oeis.org

8634368000, 175898504162692612600853299200000, 569584588357708246352221171200000000000, 30903356351561222538825891668691517440000, 75783579745006249731939558173442048000000, 221669803220344213547594484715842176665500, 11477043671509412495692698759458678235987968000
Offset: 1

Views

Author

David A. Madore, Nov 08 2010

Keywords

Comments

Terms in this sequence are the terms which are repeated in A121732.
This sequence is infinitely long. For example, there exists two non-isomorphic irreducible E8 representations of dimension n^120 * 8634368000 for any n>0 (the dimension formula is homogeneous of degree 120 after a suitable change of variables). I have found no dimension with more than two non-isomorphic irreducible E8 representations. - Andy Huchala, Nov 22 2020

Examples

			With the fundamental weights numbered as in Bourbaki, the highest weights 10100000 and 10000011 both correspond to irreducible representations of dimension 8634368000. The highest weights 23000130 and 12000231 both correspond to irreducible representations of dimension 175898504162692612600853299200000.

References

  • N. Bourbaki, Lie Groups and Lie Algebras Chapters 4-6, Springer, 1968, 283-285.

Links

Crossrefs

Cf. A121732.

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

Views

Author

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

Keywords

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.

Examples

			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.

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

Crossrefs

Cf. A121732, A121741.

Programs

Formula

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.

A030650 Dimensions of multiples of minimal representations of complex Lie algebra E_8.

Original entry on oeis.org

248, 27000, 1763125, 79143000, 2642777280, 69176971200, 1473701482500, 26284473168750, 401283501480000, 5338265882241600, 62790857238950100, 661062273763905000, 6294003651511200000, 54675736068345120000
Offset: 1

Views

Author

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

Keywords

Comments

Dimensions of certain Lie algebra (see Landsberg-Manivel reference for precise definition). - N. J. A. Sloane, Oct 15 2007

References

  • Cf. table 5 of Seminar on Lie Groups and Algebraic Groups of Onishchik and Vinberg [ Springer Verlag 1990 ].

Links

Crossrefs

Cf. A121732.

Programs

  • Maple
    b:=binomial; t71:= proc(a,k) ((3*a+2*k+5)/(3*a+5)) * b(k+2*a+3,k)*b(k+5*a/2+3,k)*b(k+3*a+4,k)/(b(k+a/2+1,k)*b(k+a+1,k)); end; [seq(t71(8,k),k=0..30)]; # N. J. A. Sloane, Oct 15 2007

Formula

a(n) = (1/298109643686752257360)*(2*n+29)*binomial(n+28, 5)*binomial(n+19, 10)* binomial(n+5, 5)*binomial(n+23, 18)^2.

A340522 Weight multiplicities for the trivial representation of the Lie algebra E_8.

Original entry on oeis.org

1, 8, 35, 140, 120, 370, 1407, 2960, 4480, 1765, 6000, 4104, 18688, 43065, 37680, 64470
Offset: 0

Views

Author

N. J. A. Sloane, Feb 08 2021

Keywords

Links

Crossrefs

Cf. A004009, A121732, A340523, A341289, A341290.
Showing 1-10 of 15 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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