Unitary matrix integrals over symmetric polynomials play an important role in a wide variety of applications, including random theory, gauge number and enumerative combinatorics. We derive novel results on such apply these other identities to correlation functions long-range walks (LRRW) consisting hard-core bosons. generalize identity due Diaconis Shahshahani which computes unitary products po...

in this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric connes amenability. we determine symmetric module amenability and symmetric connes amenability of some concrete banach algebras. indeed, it is shown that $ell^1(s)$ is  a symmetric $ell^1(e)$-module amenable if and only if $s$ is amenable, where $s$ is an inverse semigroup with subsemigr...