OFFSET
0,5
COMMENTS
If A is a random matrix in the compact group USp(6) (6 X 6 complex matrices which are unitary and symplectic), then a(n) = E[(tr(A))^n] is the n-th moment of the trace of A.
The multiplicity of the trivial representation in the n-th tensor power of the standard representation of USp(6).
Number of returning walks of length n on a cubic lattice remaining in the chamber x >= y >= z >= 0.
Under a generalized Sato-Tate conjecture, this is the moment sequence of the distribution of unitarized Frobenius traces a_p/sqrt(p) (as p varies), for almost all genus 3 curves.
For genus g the mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=g, where F_m(z) = Sum_j binomial(m,j)(I_{2j-m}(2z)-I_{2j-m+2}) and I_k(z) is the hyperbolic Bessel function (of the first kind) of order k.
Dimension of space of invariant tensors in n-th tensor power of natural representation of Sp(6). - Bruce Westbury, Dec 05 2014
LINKS
David J. Grabiner and Peter Magyar, Random walks in Weyl chambers and the decomposition of tensor powers, Journal of Algebraic Combinatorics, vol. 2 (1993), no. 3, pp 239-260.
Nicholas M. Katz and Peter Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy, AMS, 1999.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
G. Lachaud, On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius, arXiv preprint arXiv:1506.06482 [math.AG], 2015.
FORMULA
mgf: A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j binomial(m,j)(I_{2j-m}(2z)-I_{2j-m+2}(2z)) and I_k(z) is the hyperbolic Bessel function (of the first kind) of order k.
EXAMPLE
a(4)=3 because E[(tr(A)^4] = 3 for a random matrix A in USp(6).
MATHEMATICA
F[m_][z_] := Sum[Binomial[m, j] (BesselI[2j-m, 2z] - BesselI[2j-m+2, 2z]), {j, 0, m}];
A[z_] := Det[Table[F[i+j-2][z], {i, 1, 3}, {j, 1, 3}]];
a[n_] := a[n] = Derivative[n][A][0];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 17 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew V. Sutherland, Mar 24 2008, Apr 01 2008
STATUS
approved