A138540 Moment sequence of tr(A) in USp(6).

1, 0, 1, 0, 3, 0, 15, 0, 104, 0, 909, 0, 9449, 0, 112398, 0, 1489410, 0, 21562086, 0, 336086022, 0, 5577242292, 0, 97671172836, 0, 1792348213025, 0, 34268124834495, 0, 679376016769260, 0, 13911118850603610, 0, 293220749128031010, 0

Offset: 0

Author

Andrew V. Sutherland, Mar 24 2008, Apr 01 2008

Keywords

• nonn

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

Examples

			a(4)=3 because E[(tr(A))^4] = 3 for a random matrix A in USp(6).
		

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.

Crossrefs

Cf. A138349.

Programs

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 *)

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.