The permanent functions of tensors

TL;DR

This paper explores the definitions, properties, and extensions of permanent functions for tensors, including their geometric interpretations and the structure of related polytopes, advancing the theoretical understanding of tensor permanents.

Contribution

It introduces new definitions of tensor permanents, discusses the polytopes of stochastic tensors, and extends the generalized matrix function to tensors, enriching tensor algebra theory.

Findings

Definitions of 1-, 2-, and k-permanent for tensors

Characterization of polytopes of stochastic tensors

Extension of the generalized matrix function to tensors

Abstract

By a tensor we mean a multidimensional array (matrix) or hypermatrix over a number field. This article aims to set an account of the studies on the permanent functions of tensors. We formulate the definitions of 1-permanent, 2-permanent, and $k$-permanent of a tensor in terms of hyperplanes, planes and $k$-planes of the tensor; we discuss the polytopes of stochastic tensors; at end we present an extension of the generalized matrix function for tensors.