On the representations of the full matrix semigroup on homogeneous polynomials. II (Q1073190)

This paper is a continuation of the author's paper on the same subject [cf. the preceding review]. With the same notations as there the author describes the structure of W in the singular case: The author proves: (1) For every partition \(\lambda =(r,q)\), \(q\leq r\leq p-2\) the module \(S^{\lambda}\uparrow {\mathcal S}_{| \lambda | +1}\) is a distributive \({\mathcal S}_ n\)-module. (2) \(W^{(d)}\) is a D-module over \(M_ n\), \(n\geq d\) if and only if d is as follows: (i) If \(p>5\), then \(d<p^ 2\) and when d is written to the base p, \(d=a+bp\) we have (a) \(b\leq a+4\) and (b) \(a+b\leq p+1\). (ii) If \(p=5\), then add \(d=28\) to the condition on d given in (i). (iii) If \(p=3\), then add \(d=10,11\) to the conditions given in (i). (iv) If \(p=2\), then \(d<6\).