On maximally symmetric subalgebras

Abstract Let $$\mathbb {k}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> be a characteristic zero Dedekind domain, S be a $$\mathbb {k}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -algebra, and $$T\subseteq S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> be a full rank subalgebra. Suppose the algebra T is symmetric. It is important to know when T is a maximally symmetric subalgebra of S , i.e., no $$\mathbb {k}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -subalgebra C satisfying $$T\subsetneq C\subseteq S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>⊊</mml:mo> <mml:mi>C</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> is symmetric. In this note, we establish a useful sufficient condition for this using a notion of a quasi-unit of an algebra. This condition is used to obtain an old and a new result on maximal symmetricity for generalized Schur algebras corresponding to certain Brauer tree algebras. The old result was used in our work with Evseev on RoCK blocks of symmetric groups. The new result will be used in our forthcoming work on RoCK blocks of double covers of symmetric groups.