Darij Grinberg: The Petrie symmetrie functions and Murnaghan-Nakayama rules

Given integers \(k > 0\) and \(m \geq 0\), we define the symmetric function \(G(k, m)\) to be the sum of all degree-\(m\) monomials in countably many variables \(x_1, x_2, x_3, \ldots\) that contain no exponent \(\geq k\). This function \(G(k, m)\), which I have taken to calling a "Petrie symmetric function", originated in a proof of a conjecture by Liu and Polo from the theory of algebraic groups; I will discuss its combinatorial properties. Perhaps the most striking one is a Murnaghan-Nakayama-like rule for expanding the product \(G(k, m) s_\lambda\) in the Schur basis, where \(s_\lambda\) is a Schur function. The coefficients in this expansion are \(0\)'s, \(1\)'s and \(-1\)'s, and can be described as determinants of Petrie matrices. In the case when \(\lambda = \varnothing\) (so we are expanding \(G(k, m)\) itself), they can also be described in a purely combinatorial way. I will discuss this and further properties (including new bases constructed of the \(G(k, m)\) for a fixed \(k\)), as well as the question what other symmetric functions have this Murnaghan-Nakayama-like property.