Permanents and determinants with generic noncommuting entries

Abstract

This paper studies some basic combinatorial properties of matrix functions of generic matrices. A generic matrix is one with entries from a free associative algebra, over a field, and on a finite set of non-commuting variables (i.e. a tensor algebra). The principal tools are shuffle products. Generic column and row permanents are defined and analogs of the Laplace and Cauchy-Binet theorems are derived in terms of shuffles. In this setting, the generic permanents include as special cases all of the classical matrix functions: Schur matrix functions, determinants, and permanents. 1980 Mathematics Classification 05, 15. Keywords: Shuffle product, generic matrix functions, minor expansions, Laplace Expansion Theorem, Cauchy-Binet Theorem, permanents, determinants, tensor algebra, matrices with non-commuting entries.