Symmetric and r-symmetric tropical polynomials and rational functions

Introductions to Symmetric Polynomials and Symmetric Functions

Symmetric polynomials and symmetric functions are ubiquitous in mathematics and mathematical physics. For example, they appear in elementary algebra (e.g. Viete’s Theorem), representation theories of symmetric groups and general linear groups over C or finite fields. They are also important objects to study in algebraic combinatorics. Via their close relations with representation theory, the th...

Symmetric Functions and Macdonald Polynomials

The ring of symmetric functions Λ, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on Λ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood–...

Symmetric Functions and Hall Polynomials

Partition functions and symmetric polynomials

We find a close correspondence between certain partition functions of ideal quantum gases and certain symmetric polynomials. Due to this correspondence it can be shown that a number of thermodynamic identities which have recently been considered are essentially of combinatorical origin and known for a long time as theorems on symmetric polynomials. For example, a recurrence relation for partiti...

Generalized Bernstein Polynomials and Symmetric Functions

We begin by classifying all solutions of two natural recurrences that Bernstein polynomials satisfy. The first scheme gives a natural characterization of Stancu polynomials. In Section 2, we identify the Bernstein polynomials as coefficients in the generating function for the elementary symmetric functions, which gives a new proof of total positivity for Bernstein polynomials, by identifying th...