# Groups, representations, and cohomology on Skye

A couple of weeks ago I attended a summer school on the Isle of Skye, organised by the Anglo-Franco-German representation theory network. The aim of the school was to understand the following theorem of Benson, Carlson, and Rickard: that thick tensor ideals of the stable module category of a finite group are classified by the specialisation closed subsets of the projective variety associated with the cohomology ring of the group. In particular we studied a new proof of this theorem due to Carlson and Iyengar.

This topic was a little way away from my usual research, but I found it enormously interesting. What is most fascinating is how much has been achieved in this field with homological algebra. I understand that this has been in part through necessity:

There is a legendary story that Brauer, himself, used to advise his students not to try to study the representation theory of p-groups. The subject seemed to be too difficult with little or no promise of productive results. Yet for the investigation of module structure, many of the most fascinating and difficult problems can be easily reduced to questions involving the representations of p-groups over fields of characteristic p. On the other hand, in this situation, all group characters are trivial, the Grothendieck group is trivial, and many of the classical techniques of representation theory have no relevance. The only method left open to us is homological algebra.1

In any case, I now have one more reason to get to work learning about triangulated and derived worlds.

Of course, one can’t go to Skye and only do maths! I travelled to the summer school by the most beautiful possible route: conventional trains to Edinburgh and from Edinburgh to Glasgow; the famous, and absolutely stunning, West Highland Line from Glasgow to Mallaig via Fort William, which skirts Loch Lomond and crosses the Glenfinnan viaduct; finally, the breezy ferry crossing from Mallaig to Armadale. Registration for the summer school didn’t start until the day after I arrived, so I took advantage of Scotland’s freedom to roam, and wild camped the night. It was cold and I got ticks. The next day I warmed up at the Talisker distillery before heading back to register at the Gaelic college where the summer school was being hosted.

The rest of the week was quite busy, but I managed to fit in some more successful outdoorsing. On the free afternoon I went with three others to climb a nearby mountain, Beinn na Cailleach. It has an especially large summit cairn, reputedly the burial site of a Norwegian princess:

…the cairn situated on the summit of Beinn na Cailleach, not far from Broadford… This cairn is believed to mark the site of burial of a Norse princess who died at Ord. On her deathbed this princess commanded her attendants to convey her, when dead to the top of Beinn na Cailleach, and to bury her there, in order that she might lie in the wake of the winds from Norway.2

Another evening I led a small group on a shorter walk to see some waterfalls. We swam in a loch and then got lost on the way back. Luckily it is light almost until midnight in June on Skye, so we could at least see where we were going. And during a couple of lunchtimes I went down to a beach to swim (very briefly) in the sea. It was much colder than the loch!

Skye is fantastically beautiful, and I was pleased to have the opportunity to go. I’ll definitely be going back, hopefully for a holiday so I have more time to explore.

1. From the preface Modules and group algebras, J. F. Carlson. ↩︎

2. The Peat-Fire Flame, Alasdair Alpin Macgregor. ↩︎

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