On the action of the Iwahori–Hecke algebra on modular invariants

Abstract

We compute the action of the modular Iwahori–Hecke algebra on the ring of invariants of the mod p cohomology of elementary p-groups under Borel subgroup of the general linear group. 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 the Takayasu modules.

Keywords:

• Iwahori–Hecke algebra
• modular invariants
• Steinberg summand
• Takayasu spectra

1991 Mathematics Subject Classification:

• Primary 54C40
• 14E20
• Secondary 46E25
• 20C20

Acknowledgements

We would like to thank Nguyen D. H. Hai and Geoffrey Powell for illuminating discussions about the Iwahori–Hecke algebra and the Universal Steenrod algebra. Their notes [22] and [8] on the subject have been influential in our approach. This work was completed while the authors were visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM).

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2014.38.