On the action of the Iwahori-Hecke algebra on modular invariants. (English) Zbl 1457.20006
Let \(p\) be an odd prime, and let \(V_n\) denote an \(n\)-dimensional vector space over the prime field \(\mathbb{F}_p\). The general linear group \(G_n=\mbox{Aut}(V_n)\) acts naturally on \(V_n\) and therefore on \(H^\ast(BV_n,\mathbb{F}_p)\). Let \(B_n\) and \(U_n\) denote the Borel subgroup of \(G_n\) and write \(1^{G_n}_{B_n}\) for the representation of \(G_n\) induced by the trivial left representation of \(B_n\). The \(\mathbb{F}_p\)-algebra \(\mathcal{H}_n=\mbox{End}_{\mathbb{F}_p[G_n]}(1^{G_n}_{B_n})\) is called the \((\bmod\,p)\)-Iwahori-Hecke algebra
[N. Iwahori, J. Fac. Sci., Univ. Tokyo, Sect. I 10, 215–236 (1964; Zbl 0135.07101)]. For any left \(G_n\)-module \(M\), the invariant subspace \(M^{B_n}\) is equipped with a natural (right) action of \(\mathcal{H}_n\).
The authors compute explicitly this action for the \(G_n\)-module \(H^\ast(BV_n,\mathbb{F}_p)\) at an odd prime. Applications include a direct proof of the structure of the universal Steenrod algebra and a new proof of a key result on the structure of Takayasu modules [S.-i. Takayasu, J. Math. Kyoto Univ. 39, No. 2, 377–398 (1999; Zbl 1002.55006)].
The authors compute explicitly this action for the \(G_n\)-module \(H^\ast(BV_n,\mathbb{F}_p)\) at an odd prime. Applications include a direct proof of the structure of the universal Steenrod algebra and a new proof of a key result on the structure of Takayasu modules [S.-i. Takayasu, J. Math. Kyoto Univ. 39, No. 2, 377–398 (1999; Zbl 1002.55006)].
Reviewer: Marek Golasiński (Olsztyn)
MSC:
| 20C08 | Hecke algebras and their representations |
| 20C20 | Modular representations and characters |
| 55S10 | Steenrod algebra |
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