| Abstract: | We define the algebrant, a mathematical generalization of the determinant, the immanant, the permanent, and the Schur functions. Algebrants are classified as multilinear matrix functions or multicomponent symmetrized tensors. In applications, such as N-electron quantum mechanics, where extensive computation is required, it is vital to reduce computational effort, e.g., the well-known N-factorial problem. We derive certain mathematical properties that can be incorporated in efficient computing algorithms for algebrants. Foremost is our “elimination theorem,” which allows (in important special cases) zeros to be introduced into an algebrant in close analogy with Gaussian elimination for determinants. Savings accruing from such elimination can be substantial. We show examples from Matsen's spin-free quantum chemistry where elimination effectively removes the N-factorial problem that has hitherto stifled possible applications. |
