A181746 -id:A181746 - cp's OEIS frontend

A339248 List of dimensions for which there exist several non-isomorphic irreducible representations of G2.

77, 2079, 4928, 30107, 56133, 133056, 315392, 812889, 1203125, 1515591, 1926848, 3592512, 8515584, 9058973, 20185088, 21948003, 32484375, 40920957, 52024896, 77000000, 96997824, 123318272, 136410197, 229920768, 244592271, 342513171, 371664293, 470421875

Offset: 1

Andy Huchala, Nov 28 2020

nonn

Terms which could be repeated in A104599.

There are infinitely many terms in this sequence as the dimension formula is homogeneous of degree 6; see A181746.

			With the fundamental weights numbered as in Bourbaki, the highest weights 3,0 and 0,2 both correspond to irreducible representations of dimension 77. The highest weights 2,3 and 8,0 both correspond to irreducible representations of dimension 2079.

N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6, Springer, 1968, 231-233.

Andy Huchala, Table of n, a(n) for n = 1..20000
Andy Huchala, C++ program
Wikipedia, G2 (mathematics)

Cf. A181746, A104599.

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 and duplicates recorded.

A339249 List of dimensions for which there exist several non-isomorphic irreducible representations of F4.

Original entry on oeis.org

1053, 160056, 4313088, 28481544, 655589376, 17666408448, 30011240259, 116660404224, 168658209720, 257425688520, 2585493646164, 2685294084096, 7548492087864, 9283085543160, 9283085543160, 32912757834840, 62027889765660, 72361609003008, 81736097625000

Offset: 1

Andy Huchala, Nov 28 2020

nonn

Terms in this sequence are the terms which could be repeated in A121738.

This sequence is infinitely long; see A181746.

			With the fundamental weights numbered as in Bourbaki, the highest weights 1001 and 2000 both correspond to irreducible representations of dimension 1053. The highest weights 0102 and 2002 both correspond to irreducible representations of dimension 160056.

N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6, Springer, 1968, 223-224.

Andy Huchala, Table of n, a(n) for n = 1..804
Andy Huchala, C++ program

Cf. A121738, A181746.

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 and duplicates recorded.

A339250 List of dimensions for which there exist several non-isomorphic irreducible representations of E6.

Original entry on oeis.org

27, 351, 1728, 3003, 5824, 7371, 7722, 17550, 19305, 34398, 46332, 51975, 54054, 61425, 78975, 100386, 112320, 146432, 252252, 314496, 359424, 371800, 386100, 393822, 412776, 442442, 459459

Offset: 1

Andy Huchala, Apr 02 2021

nonn

Terms which would be repeated in A121737.
There are infinitely many terms in this sequence; see A181746.
By symmetry of the Dynkin diagram, with fundamental weights numbered as in Bourbaki there is a duality of highest weights [1,0,0,0,0,0] and [0,0,0,0,0,1]. Similarly, there is a duality of highest weights [0,0,0,0,1,0] and [0,0,1,0,0,0]. Note that E6 is the only exceptional Lie algebra with such a duality. However this duality is not responsible for all pairs of non-isomorphic irreducible E6 representations of equal dimension--see example.
There are 6 non-isomorphic irreducible E6 representations of dimension 7183313280, and 8 non-isomorphic irreducible E6 representations of dimension 7980534952482277785600. Both dimensions are minimal with respect to that property. I do not know if such dimensions exist for 9 or more irreducible representations.

			With the fundamental weights numbered as in Bourbaki, the irreducible E6-modules with highest weights [1,0,0,0,0,0] and [0,0,0,0,0,1] both have dimension 77. The vectors [0,0,0,0,1,0], [0,0,1,0,0,0], [2,0,0,0,0,0], and [0,0,0,0,0,2] are the four highest weights which correspond to irreducible representations of dimension 351.

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, E6 (mathematics)

Cf. A181746, A121737.

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

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.

A343266 List of dimensions for which there exist 8 or more non-isomorphic irreducible representations of E6.

Original entry on oeis.org

7980534952482277785600, 81594430454916707328000, 14562708974765215732435968000, 548418186007940753739828795801600, 5607126565433818566044216721408000, 1000741740604516653587154703585640448000, 1197835483167781694195253526401026457600

Offset: 1

Andy Huchala, Apr 09 2021

nonn

Terms which would appear in A121737 eight times or more if it included repeats.

There are infinitely many terms in this sequence; see A181746. For example, 548418186007940753739828795801600 = 2^36 * 7980534952482277785600, and additional dimensions may be obtained by replacing 2^36 with n^36 for any positive integer n.

			With fundamental weights as ordered in Bourbaki, the eight irreducible E6 representations with dimension 7980534952482277785600 have highest weights as follows:
  [ 1, 5, 2, 5, 0, 7]
  [ 7, 5, 0, 5, 2, 1]
  [ 5, 7, 2, 3, 2, 1]
  [ 1, 7, 2, 3, 2, 5]
  [ 1, 5, 6, 1, 0,11]
  [11, 5, 0, 1, 6, 1]
  [ 4, 5, 3, 1, 0,14]
  [14, 5, 0, 1, 3, 4]

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..12
Andy Huchala, C++ Program

Cf. A181746, A121737.

// See A181746 and A121737.
(C++) // See A181746.

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A339248 List of dimensions for which there exist several non-isomorphic irreducible representations of G2.

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A339249 List of dimensions for which there exist several non-isomorphic irreducible representations of F4.

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A339250 List of dimensions for which there exist several non-isomorphic irreducible representations of E6.

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

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Programs

Java

A343266 List of dimensions for which there exist 8 or more non-isomorphic irreducible representations of E6.

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Java