On matrix representations of groups of order 𝑝 ⁵ over ℚ

In this article, we study rational representations of groups of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mn>5</m:mn> </m:msup> </m:math> p⁵ , where 𝑝 is an odd prime. For a 𝑝-group 𝐺 and an irreducible complex character 𝜒 of 𝐺, the construction of an irreducible rational matrix representation of 𝐺 affording the character <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Ω(χ) is equivalent to determining a pair <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>H</m:mi> <m:mo>,</m:mo> <m:mi>ψ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (H,ψ) , with 𝐻 a subgroup of 𝐺 and 𝜓 a linear character of 𝐻 such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>ψ</m:mi> <m:mi>G</m:mi> </m:msup> <m:mo>=</m:mo> <m:mi>χ</m:mi> </m:mrow> </m:math> ψG=χ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ψ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> Q(ψ)=Q(χ) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>m</m:mi> <m:mi mathvariant="double-struck">Q</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:msub> <m:mo>∑</m:mo> <m:mrow> <m:mi>σ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>Gal</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>/</m:mo> <m:mi mathvariant="double-struck">Q</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:msub> <m:msup> <m:mi>χ</m:mi> <m:mi>σ</m:mi> </m:msup> </m:mrow> </m:mrow> </m:mrow> </m:math> Ω(χ)=mQ(χ)∑σ(Q(χ)/Q)χσ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>m</m:mi> <m:mi mathvariant="double-struck">Q</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> mQ(χ) denotes the Schur index of 𝜒 over ℚ. For each inequivalent irreducib