0905.1992 (Sho Matsumoto et al.)

Jucys-Murphy Elements and Unitary Matrix Integrals    [PDF]

Sho Matsumoto, Jonathan Novak

In this paper, we study the relationship between polynomial integrals on the
unitary group and the conjugacy class expansion of symmetric functions in
Jucys-Murphy elements. Our main result is an explicit formula for the top
coefficients in the class expansion of monomial symmetric functions in
Jucys-Murphy elements, from which we recover the first order asymptotics of
polynomial integrals over $\U(N)$ as $N \rightarrow \infty$. Our results on
class expansion include an analogue of Macdonald's result for the top
connection coefficients of the class algebra, a generalization of Stanley and
Olshanski's result on the polynomiality of content statistics on
Plancherel-random partitions, and an exact formula for the multiplicity of the
class of full cycles in the expansion of a complete symmetric function in
Jucys-Murphy elements. The latter leads to a new combinatorial interpretation
of the Carlitz-Riordan central factorial numbers.

View original: http://arxiv.org/abs/0905.1992