Algorithmically distinguishing irreducible characters of the symmetric group

TL;DR

This paper presents a polynomial-time algorithm to distinguish and identify irreducible characters of the symmetric group using minimal queries, advancing computational methods in algebraic representation theory.

Contribution

It introduces an efficient algorithm for distinguishing and reconstructing irreducible characters of symmetric groups from oracle access, a novel approach in computational algebra.

Findings

Algorithm distinguishes irreducible characters in polynomial time.

Algorithm reconstructs the character label using polynomially many queries.

Queries and computations are feasible within polynomial time bounds.

Abstract

Suppose that $\chi_\lambda$ and $\chi_\mu$ are distinct irreducible characters of the symmetric group $S_n$. We give an algorithm that, in time polynomial in $n$, constructs $\pi\in S_n$ such that $\chi_\lambda(\pi)$ is provably different from $\chi_\mu(\pi)$. In fact, we show a little more. Suppose $f=\chi_\lambda$ for some irreducible character $\chi_\lambda$ of $S_n$, but we do not know $\lambda$, and we are given only oracle access to $f$. We give an algorithm that determines $\lambda$, using a number of queries to $f$ that is polynomial in $n$. Each query can be computed in time polynomial in $n$ by someone who knows $\lambda$.