Poincaré series for the occurrence of certain modular representations of GL(n,p) in the symmetric algebra
- 1 January 1989
- journal article
- research article
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- Vol. 113 (1-2) , 27-41
- https://doi.org/10.1017/s0308210500023933
Abstract
The number of occurrences of the Steinberg representation St of GL(n, p) as a composition factor in the symmetric algebra Fp[x1, … xn] has been determined by several authors. We extend this result to the representations of GL(n, p) which are the closest neighbours of St in the SL(n, p) weight diagram. The method is to play off duality for GL(n, p)-modules against connectivity for M(n, p)-modules. The result is equivalent to determining the cohomology groups of the corresponding indecomposable stable summands of the localisation of an n-fold product of complex projective spaces at the prime p.Keywords
This publication has 7 references indexed in Scilit:
- Subalgebras of the steenrod algebra and the action of matrices on truncated polynomial algebrasJournal of Algebra, 1989
- Smash products of summands of 𝐵(𝑍/𝑝)ⁿ₊Contemporary Mathematics, 1989
- On the iterated complex transferMathematische Zeitschrift, 1988
- Stable decompositions of classifying spaces of finite abelianp-groupsMathematical Proceedings of the Cambridge Philosophical Society, 1988
- Finite complexes with A(n)-free cohomologyTopology, 1985
- Modular representations ofGL(n, p), splitting ?(?P ?x...x?P ?), and the?-family as framed hypersurfacesMathematische Zeitschrift, 1985
- Stable splittings derived from the Steinberg moduleTopology, 1983