Harman and Snowden have developed a theory of pre-Tannakian categories coming from Oligomorphic permutation groups together with a well-behaved measure on G-sets. The prototypical example is Deligne's S_t which comes from the infinite symmetric group together with a measure where the usual permutation G-set has volume t. The simplest new example of their theory is the group of order-preserving bijections of the real line with the measure given by Euler characteristic. In joint work with Harman and Snowden we give a detailed description of this new symmetric tensor category, which has a number of novel properties. We call this category the Delannoy category because the dimensions of Hom spaces are given by Delannoy numbers. In this talk I'll outline our main results, including the classification of simple objects, the tensor product rules, and a combinatorial model for the category using Delannoy paths. At the end I'll briefly discuss a non-semisimple variant we call the "circular Delannoy category" coming from the group of order-preserving bijections of the circle with the measure given by Euler characteristic.
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