Group algebras, monotonicity, and the lieb permanent inequality

Abstract

If n is a positive integer then let S _n denote the symmetric group on let {1,2, …n} letCS _n denote the algebra of functions from S _nto C, and let H _n denote the n × n positive semi-definite Hermitian matrices. Define the function [.](.) from for each Fix k such that 0<k<n and associate with n × n complex matrix the complementary principal submatrices A ^[k] and ^[k] . Lieb's permanent inequality guarantees us thai if A∊H _nthen per n(a) per(A^[k]) per (A^(k))We introduce a sequence P ₀ P ₁ P ₂ … P _k of normalized positive semi-definite Hermitian members of _n such that the associated sequence [P ₀] (·), [P ₁](·), …, [P_k ](·) of generalized matrix functions is non-increasing when restricted to H_n . Since [P ₀](A) = per(A) and [P_k ](A) = per(A ^{[ k]}) per(A ^(k),) this comprises a refinement of Lieb's permanent inequality. We demonstrate that the matrix functions [P_t ](·), 0≤t≤k, are closely related to the Laplace expansion formulas for the permanent and determinant functions. For example, if A ∊ H_n and A = [a_ij ] then the inequality [P ₀](A) ≥ [P ₁](A) translates into where (A(i,j) is A with row i and column j deleted.

We present an extension of Lieb's inequality involving the higher order differences of the sequence P ₀,P ₁,P ₂,… P_k . In particular, defining the difference operator ▽ according to we show that if q + t ≤ k, then for each A ∊ H_n .