Seminar

# The Petrie symmetrie functions and Murnaghan-Nakayama rules

#### Algebraic and Enumerative Combinatorics

#### 04 February 11:00 - 11:50

#### Darij Grinberg - Drexel University

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.

Organizers

Sara Billey

University of Washington

Petter Brändén

KTH Royal Institute of Technology

Sylvie Corteel

Université Paris Diderot, Paris 7

Svante Linusson

KTH Royal Institute of Technology