Vol. 77, No. 2, 1978

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The number of nonfree components in the decomposition of symmetric powers in characteristic p

Gert Einar Torsten Almkvist

Vol. 77 (1978), No. 2, 293–301

If G is the group with p (=prime) elements and k a field of characteristic p let V 1,V 2,,V p denote the indecomposable k[G]-modules of k-dimension 1,2,,p respectively. Let en,ν denote the number of nonfree components of the decomposition of the symmetric power SνV n+1. Then the following symmetry relation is proved

en,p−n−ν−1 = en,ν.

As a corollary we find that SrV n+1 has exactly one nonfree component when n + r = p 2 thus solving a problem in a previous paper by R. Fossum and the author. An explicit formula for en,ν expressed in numbers of restricted partitions is obtained.

Mathematical Subject Classification 2000
Primary: 13F20
Secondary: 14C99, 15A72
Received: 22 September 1977
Published: 1 August 1978
Gert Einar Torsten Almkvist