A note on bideterminants for Schur superalgebras

Let S(m|n,r)Z be a Z-form of a Schur superalgebra S(m|n,r) generated by elements ξi,j. We solve a problem of Muir and describe a Z-form of a simple S(m|n,r)-module Dλ,Q over the field Q of rational numbers, under the action of S(m|n,r)Z. This Z-form is the Z-span of modified bideterminants [Tℓ:Ti] defined in this work. We also prove that each [Tℓ:Ti] is a Z-linear combination of modified bideterminants corresponding to (m|n)-semistandard tableaux Ti.