Skip Garibaldi - cp's OEIS frontend

A280191 Essential dimension of the spin group Spin_n over an algebraically closed field of characteristic different from 2.

0, 0, 4, 5, 5, 4, 5, 6, 6, 7, 23, 24, 120, 103, 341, 326, 814, 793, 1795, 1780, 3796, 3771, 7841, 7818, 15978, 15949, 32303, 32304, 65008, 64975, 130477, 130446, 261478, 261441, 523547, 523516, 1047756, 1047715, 2096249, 2096210, 4193314, 4193269, 8387527, 8387496, 16776040, 16775991

Offset: 5

Skip Garibaldi, Dec 28 2016

nonn

For n <= 14, due to Markus Rost. For n > 14, see references.

			a(14) = 7, meaning that Spin_14 has essential dimension 7, reflecting a cohomological invariant of degree 7 constructed using the G2 X G2 semidirect mu_4 subgroup.

S. Garibaldi, "Cohomological invariants: exceptional groups and spin groups", Memoirs of the AMS #937 (2009).
A. Merkurjev, Essential dimension, Quadratic forms-algebra, arithmetic, and geometry (R. Baeza, W.K. Chan, D.W. Hoffmann, and R. Schulze-Pillot, eds.), Contemp. Math., vol. 493, 2009, pp. 299-325.

P. Brosnan, Z. Reichstein, and A. Vistoli, Essential dimension, spinor groups, and quadratic forms, Annals of Math. vol 171 (2010), 533-544.
V. Chernousov and A.S. Merkurjev, Essential dimension of spinor and Clifford groups, Algebra & Number Theory 8 (2014), no. 2, 457-472.
S. Garibaldi and R.M. Guralnick, Spinors and essential dimension, arXiv:1601.00590 [math.GR], 2016.
Alexander S. Merkurjev, Essential dimension, Bull. Amer. Math. Soc., 54 (Oct. 2017), 635-661.

Agrees with sequence A163417 for n > 15 and not divisible by 4. First term of agreement is a(17) = 120.

a[n_] := If[n>14, Which[Mod[n, 2] == 1, 2^((n-1)/2)-n(n-1)/2, Mod[n, 4] == 2, 2^((n-2)/2)-n(n-1)/2, Mod[n, 4] == 0, 2^IntegerExponent[n, 2]-n(n-1)/2 + 2^((n-2)/2)], If[n >= 5, {0, 0, 4, 5, 5, 4, 5, 6, 6, 7}[[n-4]]]];
Table[a[n], {n, 5, 50}] (* Jean-François Alcover, Feb 18 2019, from Python *)

def a(n):
    if n > 14:
        if n%2 == 1:
            return 2**((n-1)/2) - n*(n-1)/2
        if n%4 == 2:
            return 2**((n-2)/2) - n*(n-1)/2
        if n%4 == 0:
            return 2**((n-2)/2) - n*(n-1)/2 + biggestdivisor(n,2)
    elif n >= 5:
        return [0,0,4,5,5,4,5,6,6,7][n-5]
    return "Error"
def biggestdivisor(n,d): # return largest power of d dividing n
    if n%d != 0:
        return 1;
    else:
        return d*biggestdivisor(n/d, d);

More terms from Jean-François Alcover, Mar 12 2019

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.

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.

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.

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.

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.

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

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.

cp's OEIS Frontend

User: Skip Garibaldi

A280191 Essential dimension of the spin group Spin_n over an algebraically closed field of characteristic different from 2.

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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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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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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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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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Examples

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Links

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Programs

GAP

Formula

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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Programs

GAP

Formula

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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