04.12.24 (Time of Applications, finishing phd, KLR vs symmetric groups)
Life continues, so I start to apply for postdoc positions. This is an extremely annoying, but necessary part of life as a mathematician it seems. To make it even worse: Both my friend, academic brother and coauthor Tony, and my other friend and postdoc at our group Ivan both apply to the same position.
Well let's make the mood a bit better by means of a sweet funfact. Today let's explain the world, what I recently explained Tony and what Jonas explained to me.
Start with the free symmetric k-linear monoidal category generated by one object, also known as alll the group algebras of all symmetric groups
Objects: 0,1,2,...
Morphisms n->n = group algebra of S_n
This category, call it C, is monoidal by taking permutations in S_n and S_m and tensoring them together to get a permutation in S_{n+m}. Take the Additive Completion and Karoubian closure of this, which is all the reps of the symmetric groups (at least over a characteristic zero field). This category is still monoidal, via Day convolution and the tensor product of two representations, one of S_n and one of S_m is given by their vector space tensor sproduct induced up to a rep of S_{n+m}. In this light the functor 1\otimes - is the functor, which induces up whatever representation of whatever S_n to the corresponding rep of S_{n+1}. This functor turns outto not be indecomposable and instead decomposes of a direct sum of Z (integers) many functurs {E_i |i\in Z}. These functors define a categorical action of the positive part of U(gl_{infty}) on all the reps. Moreover one can see that the monoidal subcategory of endofunctors of C generated by those is a quotient of the KLR algebra (=quiver Hecke algebra) for the quiver, which is one very long directed line. This explains how all the symmetric groups and their reps are just a very small special case of KLR. Madness.