Symmetric Functions Algebras (SFA) II: Induced Matrices
Philip Feinsilver · Journal of Stochastic Analysis · 2023
We study symmetric functions algebras 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.Thus, we determine corresponding elementary, homogeneous, power sum, monomial, and Schur functions and study their properties.For symmetric tensor powers, stochastic matrices are mapped to stochastic matrices with corresponding Markov chains arising from a given underlying chain.This paper details the construction of the multinomial chains and then looks at symmetric functions algebras based on the induced matrix map for a general matrix.A novel proof of the trace formula based on a multidimensional extension of the Mehler kernel formula is provided.Extended versions of the Cayley-Hamilton theorem are given as well.