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
Keywords
Examples
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.
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
- 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.
Crossrefs
Agrees with sequence A163417 for n > 15 and not divisible by 4. First term of agreement is a(17) = 120.
Programs
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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 *) -
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);
Extensions
More terms from Jean-François Alcover, Mar 12 2019
Comments