Schur Superalgebras in Characteristic p

Abstract

The structure of a Schur superalgebra S ¼ S(1 j 1, r) in odd characteristic p is completely determined. The algebra S is semisimple if and only if p does not divide r. If p divides r, then simple S-modules are one-dimensional and the quiver and relations of S can be immediately seen from its regular representation computed in this paper. Surprisingly, if p divides r, then S is neither quasi-hereditary nor cellular nor stratified, as one would expect by analogy with classical Schur algebras or Schur superalgebras in characteristc zero. (2000): 17A70 (primary), 20C30 (secondary). Mathematics Subject Classifications

Borel Subalgebras of Schur Superalgebras

“…The sole comprehensively studied (non-classical) case is the algebra S(1|1, r). As above, S(1|1, r) is stratifiable in the sense of [18] iff it is semisimple (see [19]). Probably we may expect a similar answer for the general case as well.…”

“…Let p|r. By [11,19], Λ(1|1, r) + = Λ(1|1, r) = {(i, r − i) | 0 i r}. Moreover, all simple S(r)-modules are one-dimensional.…”

“…Taking a dual algebra (see [19,Prop. 2.1]), we can show that the supermodule I i (E) has the following composition structure:…”

Representation type of Schur superalgebras

“…[MZ,Conj. 1] it is conjectured that S(m|n, d) is quasi-hereditary whenever d is coprime to p. In this section we use our previous calculations to show this is far from true.…”

A note on bideterminants for Schur superalgebras

“…In the case of characteristic zero or positive characteristic bigger than r, the category of supermodules over S(m|n, r) is again a highest weight category and Muir used certain bideterminants and results concerning (m, n)-hook partitions from [1] to describe a basis of standard S(m|n, r)-supermodules D λ . In the case of positive characteristic, the category of S(m|n, r)-supermodules is not a highest weight category; see [10].…”