Unisingular Specht modules

Abstract Let 𝐺 be a finite group and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ρ</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo stretchy="false">→</m:mo> <m:mrow> <m:mi>GL</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>V</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> \rho\colon G\to\operatorname{GL}(V) a finite-dimensional representation of 𝐺. We say that 𝜌 is unisingular if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo rspace="0em">det</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mrow> <m:mi>ρ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> \det(1-\rho(g))=0 for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> g\in G . Building on previous work in [J. Cullinan, Realizations of unisingular representations by hyperelliptic Jacobians, Comm. Algebra 51 (2023), 12, 5297–5312], we consider the symmetric groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> S_{n} and prove that certain families of Specht modules are always unisingular as well as raise new questions for future study.