Symmetric Functions Algebras (SFA) III: Stochastic and Constant Row Sum Matrices
Philip Feinsilver · Journal of Stochastic Analysis · 2023
We study the symmetric functions algebra based on the induced matrix map on the algebra of real d × d matrices.For fixed integer N > 0, the induced matrix map takes a matrix to the symmetric tensor power in degree N .It is determined by the action of the matrix on polynomials in d variables.The symmetric functions algebra has various bases which obey the identities of the standard algebra of symmetric functions.The case of symmetric tensor powers is taken up here when the initial element is a stochastic matrix, leading to the various bases consisting of nonnegative matrices with constant row sums .In this work, we determine row sums for the various bases: elementary, homogeneous, power sum, monomial, and Schur functions.The induced matrix map takes stochastic matrices to stochastic matrices so that in each degree N there is a corresponding Markov chain arising from a given stochastic matrix.With this interpretation, it is of interest to look at limit theorems for powers of the various elements that arise.