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A280191 Essential dimension of the spin group Spin_n over an algebraically closed field of characteristic different from 2. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 22 13:35 EDT 2019. Contains 328318 sequences. (Running on oeis4.)