by Dipper–D–Stoll (ﬁrst versions go back to 2008). The ﬁrst paper [DDS1] investigates an algebra Bn r,s (a, λ, δ) over an arbitrary commutative ring k. This algebra, a “deformation” of the walled Brauer algebra, is deﬁned by generators and relations. Bn r,s (a, λ, δ) was ﬁrst studied in Rob Leduc’s dissertation (Madison 1994) over k = C; see also Kosuda–Murakami (Osaka Math J. 1993). [DDS1] gives a description of Bn r,s (a, λ, δ) in terms of oriented tangles in the sense of Kauﬀman (TAMS 1990), and ﬁnds a basis consisting of oriented tangles, which is in bijection with the basis of (r, s)-diagrams of the walled Brauer algebra Br,s(δ). The paper ﬁnishes by proving one-half of the following result, the other half of which is proved in [DDS2]. S.R. Doty (Loyola University Chicago) Rational Schur Algebras April 13, 2013 14 / 13