Borel Subalgebras of Schur Superalgebras

Abstract

It is proved that any Schur superalgebra is representable as a product of two Borel subalgebras of that superalgebra, which are symmetric w.r.t. its natural anti-isomorphism (Bruhat-Tits decomposition). This readily implies that any simple module is uniquely defined by its highest weight, and all other weights are strictly less than is the highest under the dominant ordering. It is stated that the fundamental theorem of Kempf, which is valid for all classical Schur algebras, might be true for superalgebras only if they are semisimple. Nevertheless, a weaker theorem of Grothendieck holds true for superalgebras since Borel subalgebras are quasihereditary. Also we formulate an analog of the Donkin-Mathieu theorem for Schur superalgebras, and show that it is valid in the elementary non-classical case, that is, for the algebras S(1|1, r).

Some properties of general linear supergroups and of Schur superalgebras

“…Clearly, x π is invertible in both superalgebras K[B − ] and K[B + ], and β| A(m|n) is injective by [11,Thm. 2.1].…”

“…In [11], we defined S = S(m|n, r)-supermodules (λ) and (λ) for all λ ∈ X(T ), |λ| = r, with nonnegative coordinates. If we denote by L S (λ) a simple S-supermodule with an even highest vector of weight λ, then the first of the S-supermodules may be defined to be the greatest (super)submodule for an injective hull of L S (λ) (in the category S-mod), all weights of whom do not exceed λ.…”

“…The set of weights λ for which L S (λ) is non-trivial is denoted by X(T ) ++ . A complete description for X(T ) ++ was obtained in [20]; see also [11,16]. and since d 1 (as well as d 2 ) is not a zero divisor in K[G], the map x ⊗ d ps 1 → xd ps 1 yields the desired isomorphism.…”

“…This does in fact mean that the category in question is a highest weight one. A similar theory -with Borel supersubgroups replaced by Borel supersubalgebras -can be constructed over any Schur superalgebra (see [11]). In correspondence with induced supermodules and Weyl supermodules, in this event, are costandard supermodules and standard supermodules, respectively.…”

“…It turns out that a direct analog of this theorem is invalid in the 'super' case, but if we replace one of the tensor factors by an injective supermodule over an arbitrary Schur superalgebra then an appropriate tensor product (in the category of supermodules) will have a good filtration. In [11], it was conjectured that an analog of the Donkin-Mathieu theorem would be valid for Schur superalgebras if supermodules with good filtration are replaced by polynomial supermodules with a filtration all of whose factors are costandard supermodules. There, this conjecture was verified for one important partial case.…”