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

5,3

Comments

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

References

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.

Links

Table of n, a(n) for n=5..50.

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.

Example

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.

Mathematica

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 *)

Prog

(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);

Crossrefs

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

Sequence in context: A197136 A320475 A106626 * A222703 A222587 A222378

Adjacent sequences: A280188 A280189 A280190 * A280192 A280193 A280194

Keyword

nonn

Author

Skip Garibaldi, Dec 28 2016

Extensions

More terms from Jean-François Alcover, Mar 12 2019

Status

approved