Minimal free resolutions of determinantal ideals and irreducible representations of the Lie superalgebra \(gl(m\mid n)\) (Q1375369)

Let \(S=K[x_{ij}\mid 1\leq i\leq m, 1\leq j\leq n]\) be the polynomial algebra in \(mn\) variables over a field \(K\) of characteristic \(0\). Considering \(S\) as the coordinate ring of the vector space of \(m\times n\) matrices with entries in \(K\), let \(I_p\) be the homogeneous ideal of \(S\) generated by the \(p\times p\) minors of the \(m\times n\) matrix \((x_{ij})\). The group \(GL_m(K)\times GL_n(K)\) acts naturally on \(S\) and the ideals \(I_p\) are the only prime ideals invariant under this action. The main purpose of the paper under review is to use the above group action and minimal \(S\)-free resolutions of the ideals \(I_p\) to construct some atypical irreducible representations of the Lie superalgebra \(sl(m| n)\) from the Koszul duals of the ideals \(I_p\). The authors also discuss the problem of finding minimal presentations of other atypical simple \(gl(m| n)\)-modules and extending presentations to resolutions. Techniques of the paper under review are further developed in a forthcoming paper by the same authors and yield character formulas for the modules under consideration confirming in these special cases the general character formulas for all atypical \(sl(m| n)\)-modules conjectured in [\textit{J. Van der Jeugt, J. W. B. Hughes, R. C. King} and \textit{J. Thierry-Mieg}, J. Math. Phys. 31, 2278-2304 (1990; Zbl 0725.17004)].