{ "cells": [ { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "This worksheet is an implementation of the product rules that are in\n", "\"Products of characters of the symmetric group\" (arXiv:1709.08098)\n", "by Rosa Orellana and Mike Zabrocki

\n", "Abstract: In a recent paper (arXiv:1605.06672), the authors introduced a new basis of the ring of symmetric functions which evaluate to the irreducible characters of the symmetric group at roots of unity. The structure coefficients for this new basis are the stable Kronecker coefficients. In this paper we give combinatorial descriptions for several products that have as consequences several versions of the Pieri rule for this new basis of symmetric functions. In addition, we give several applications of the products studied in this paper.

\n", "In this worksheet we will compute a few examples of the three main theorems of the paper. These theorems provide a combinatorial rule for the product of characters of the symmetric group as symmetric functions in terms of certain types of multiset tableaux. The coefficients represent the multiplicity of the irreducibles a tensor product of symmetric group modules.

\n", "This worksheet is available in both
pdf and
jupyter notebook form.
\n", "Note that the majority of the tableaux and multiset partition code that makes these functions run has not been integrated into Sage, but for the time being I am making these programs available on my website (subject to update and change):
\n", "http://garsia.math.yorku.ca/~zabrocki/msp.py
\n", "http://garsia.math.yorku.ca/~zabrocki/mst.py
\n", "Programs for calculating with the symmetric functions ${\\tilde s}_\\lambda$ and ${\\tilde h}_\\lambda$ is part of SymmetricFunctions code as a part of Sage.