%I #25 Mar 12 2019 04:07:20

%S 0,0,4,5,5,4,5,6,6,7,23,24,120,103,341,326,814,793,1795,1780,3796,

%T 3771,7841,7818,15978,15949,32303,32304,65008,64975,130477,130446,

%U 261478,261441,523547,523516,1047756,1047715,2096249,2096210,4193314,4193269,8387527,8387496,16776040,16775991

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

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

%D S. Garibaldi, "Cohomological invariants: exceptional groups and spin groups", Memoirs of the AMS #937 (2009).

%D 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.

%H P. Brosnan, Z. Reichstein, and A. Vistoli, <a href="http://dx.doi.org/10.4007/annals.2010.171.533">Essential dimension, spinor groups, and quadratic forms</a>, Annals of Math. vol 171 (2010), 533-544.

%H V. Chernousov and A.S. Merkurjev, <a href="http://www.math.ucla.edu/~merkurev/papers/i3.pdf">Essential dimension of spinor and Clifford groups</a>, Algebra & Number Theory 8 (2014), no. 2, 457-472.

%H S. Garibaldi and R.M. Guralnick, <a href="https://arxiv.org/abs/1601.00590">Spinors and essential dimension</a>, arXiv:1601.00590 [math.GR], 2016.

%H Alexander S. Merkurjev, <a href="https://doi.org/10.1090/bull/1564">Essential dimension</a>, Bull. Amer. Math. Soc., 54 (Oct. 2017), 635-661.

%e 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.

%t 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]]]];

%t Table[a[n], {n, 5, 50}] (* _Jean-François Alcover_, Feb 18 2019, from Python *)

%o (Python)

%o def a(n):

%o if n > 14:

%o if n%2 == 1:

%o return 2**((n-1)/2) - n*(n-1)/2

%o if n%4 == 2:

%o return 2**((n-2)/2) - n*(n-1)/2

%o if n%4 == 0:

%o return 2**((n-2)/2) - n*(n-1)/2 + biggestdivisor(n,2)

%o elif n >= 5:

%o return [0,0,4,5,5,4,5,6,6,7][n-5]

%o return "Error"

%o def biggestdivisor(n,d): # return largest power of d dividing n

%o if n%d != 0:

%o return 1;

%o else:

%o return d*biggestdivisor(n/d, d);

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

%K nonn

%O 5,3

%A _Skip Garibaldi_, Dec 28 2016

%E More terms from _Jean-François Alcover_, Mar 12 2019