Permanent and determinant

Generalized matrix functions, determinant and permanent

In this paper, using permutation matrices or symmetric matrices, necessary and sufficient conditions are given for a generalized matrix function to be the determinant or the permanent. We prove that a generalized matrix function is the determinant or the permanent if and only if it preserves the product of symmetric permutation matrices. Also we show that a generalized matrix function is the de...

Determinant versus permanent

Determinant Versus Permanent

We study the problem of expressing permanents of matrices as determinants of (possibly larger) matrices. This problem has close connections to the complexity of arithmetic computations: the complexities of computing permanent and determinant roughly correspond to arithmetic versions of the classes NP and P respectively. We survey known results about their relative complexity and describe two re...

Permanent versus determinant, obstructions, and Kronecker coefficients

We give an introduction to some of the recent ideas that go under the name “geometric complexity theory”. We first sketch the proof of the known upper and lower bounds for the determinantal complexity of the permanent. We then introduce the concept of a representation theoretic obstruction, which has close links to algebraic combinatorics, and we explain some of the insights gained so far. In p...

Lecture 12 – The permanent and the determinant

where (−1)σ is +1 for even permutations and −1 for odd permutations. A permutation is even if it can be obtained from the identity permutation using an even number of transpositions (where a transposition is a swap of two elements), and odd otherwise. For those more familiar with the inductive definition of the determinant, obtained by developing the determinant by the first row of the matrix, ...