Modular representations of the supergroup \(Q(n)\). II.
This is the second of two articles investigating the representation theory of the supergroup \(G=Q(n)\) over an algebraically closed field of characteristic \(p>2\). In contrast to the first article [\textit{J. Brundan, A. Kleshchev}, J. Algebra 260, No. 1, 64-98 (2003; Zbl 1027.17004)] the present article is concerned with results that in some way depend on the interpretation of the induction from a Borel subgroup \(B\) in terms of sheaf cohomology on the flag variety \(G/B\). The irreducible representations of \(G\) are parametrized by certain vectors \(\lambda\) with integer coordinates. The corresponding representation \(L(\lambda)\) can be realized as the unique irreducible submodule of the module \(H^0(\lambda)=H^0(G/B,\mathcal L(\mathfrak u(\lambda)))\), where \(\mathfrak u(\lambda)\) is a certain irreducible \(B\)-supermodule. The first main result of the paper states that the Euler characteristic \(E_\lambda\) of the characters of all higher cohomology modules \(H^i(\lambda)\) is equal to Schur's \(P\)-function, scaled by \(\dim\mathfrak u(\lambda)\). The second main result establishes the linkage principle for the Euler characteristics. Namely, it is shown that if one writes \(E_\lambda\) as the linear combination of characters of irreducible representations, then if the representation \(L(\mu)\) appears in this linear combination with a non-zero coefficient, then \(\mu\) and \(\lambda\) have the same residue content. The coefficients above are called the decomposition numbers. At the end of the paper one finds decomposition matrices for the polynomial representations of \(Q(n)\) of degree \(d\leq 12\) in characteristics \(3\) (with \(n\) large).
scientific article; zbMATH DE number 5125698