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### Session S10 - Categorification, Higher Representation Theory, and Homological Knot Invariants

Tuesday, July 20, 19:35 ~ 20:10 UTC-3

## Generators and relations for $\text{Rep}(Sp_{2n})$

### Elijah Bodish

#### University of Oregon, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak4beff7001a051615c39c65d9a0b6d907').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy4beff7001a051615c39c65d9a0b6d907 = '&#101;b&#111;d&#105;sh' + '&#64;'; addy4beff7001a051615c39c65d9a0b6d907 = addy4beff7001a051615c39c65d9a0b6d907 + '&#117;&#111;r&#101;g&#111;n' + '&#46;' + '&#101;d&#117;'; var addy_text4beff7001a051615c39c65d9a0b6d907 = '&#101;b&#111;d&#105;sh' + '&#64;' + '&#117;&#111;r&#101;g&#111;n' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak4beff7001a051615c39c65d9a0b6d907').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy4beff7001a051615c39c65d9a0b6d907 + '\'>'+addy_text4beff7001a051615c39c65d9a0b6d907+'<\/a>';

Let $\text{Rep}(G)$ be the monoidal category of finite dimensional representations of a semisimple Lie group. Kuperberg’s 1996 paper “Spiders for rank 2 Lie algebras” proposed studying $\text{Rep}(G)$ by first studying $\text{Fund}(G)$, the full monoidal subcategory generated by finite dimensional irreducible modules with highest weight a fundamental weight. Kuperberg went on to give generators and relations for $\text{Fund}(G)$ when $G = SL_3$, $Spin_5$, $Sp_4$, and $G_2$. The problem of giving analogous generators and relations for$\text{Fund}(SL_n)$ was solved in 2012 by Cautis—Kamnitzer—Morrison.

In joint work with Elias, Rose, and Tatham (https://arxiv.org/abs/2103.14997) we define a category by generators and relations and argue there is functor from the generators and relations category to $\text{Fund}(Sp_{2n})$. Combining skein theoretic arguments with combinatorial results due to Sundaram, as well as exploiting a well known relation between BMW algebras and symplectic groups, we deduce the functor is an equivalence of monoidal categories, solving Kuperberg’s problem for $\text{Fund}(Sp_{2n})$.

In the talk I will begin with an expanded discussion of the history outlined above. Then state our results and try to give an idea of how some of the arguments work by illustrating them in the case of $Sp_4$ and $Sp_6$.

Joint work with Ben Elias (University of Oregon), David Rose (UNC-Chapel Hill) and Logan Tatham (UNC-Chapel Hill).