Slides for this presentation (as well as the master's thesis it was based on) can be found at

The representation theory for the symmetric group $\mathfrak{S}_n$ over $\mathbb{C}$ is
completely controlled by its irreducible representations. These are in bijection with partitions of $n$.

The connection is cycle type: e.g. $\sigma=(1\,5\,7\,2\,6)(9\,3\,8\,4)(10)\in\mathfrak{S}_{10}$ is of the form above.

Given a representation $\rho:\operatorname{GL}_n\to \operatorname{GL}_m$, we say that
$\rho$ is *polynomial* if (there exist bases such that) the maps are polynomials
in the input coordinates.

Given a representation $\rho:\operatorname{GL}_n\to \operatorname{GL}_m$

One can use the usual notion of homogeneity to define the
*homogeneous degree $d$ polynomial representations of $\operatorname{GL}_n$.*

We denote by $P_k(n,d)=P(n,d)$ the category of homogeneous degree $d$ polynomial representations of $\operatorname{GL}_n$ over $k$.

- The module category for the Schur algebra $S(n,d)$
- Dual of a subcoalgebra of the ring of functions on the group scheme $\operatorname{GL}_n$
- Strict polynomial functors (Friedlander-Suslin)
- Objects are endo"functors" on $\operatorname{Vect}_k,$ such that the maps of morphisms are (homogeneous degree $d$) polynomial maps.
- Strict polynomial functors (Krause)
- Let $\Gamma^d P_k$ be the category of vector spaces with $\operatorname{Hom}_{\Gamma^d P_k}(V,W)=\Gamma^d\operatorname{Hom}_k(V,W)$. Then $P(n,d)\simeq \operatorname{Rep}(\Gamma^dP_k)=\operatorname{Func}(\Gamma^d P_k,\operatorname{Vect}_k).$

One can develop a theory of *weights* for $P(n,d)$ and for $d\le n$, there is a particular weight $\omega$
such that taking the $\omega$ weight space induces a functor
$$\mathcal{F}:P(n,d)\to\operatorname{Rep}(\mathfrak{S}_d)\quad\text{via}\quad \mathcal{F}(M)=M^\omega$$
along with an opposing functor
$$\mathcal G:\operatorname{Rep}(\mathfrak{S}_d)\to P(n,d),$$
both of which preserve simples.

If $k$ is any infinite field of characteristic 0 or $p>d$, we are in the semisimple case and the representation theory is mirrored in these two categories $P_k(n,d)$ and $\operatorname{Rep}_k(\mathfrak{S}_d)$.

In the modular case, things get more interesting....

How do we have a chance of studying the category of representations of something that has complicated representation type (i.e. lots of non-isomorphic indecomposables)?

Use a coarser notion of similarity than isomorphism type.

Objects like DVRs and Dedekind domains are nice to study because their spectra have nice structure.

We don't need to compute things on the individual elements of these rings to say strong things about them.

If our representation category (or a suitable analog) is "enough like a ring" we can compute the Balmer spectrum of prime (thick tensor) ideals.

It ends up that being "enough like a ring" can be justifiably interpreted as being a tensor triangulated category (TTC).

There are several places these arise in nature.

- The stable homotopy category with smash product
- The stable module category $\operatorname{stab}(kG)$
- The bounded derived category $\operatorname{D}^\mathrm{b}(A)$ for an algebra $A$ (with derived $\otimes$)

- [Balmer, Thomason] If $X$ is a topologically Noetherian scheme, then $$\operatorname{Spec}_\mathrm{Bal}\mathrm{D}^\mathrm{perf}(\mathrm{coh}(X))\cong X$$ as schemes.
- [Balmer, Friedlander-Pevtsova] If $G$ is a finite group scheme over $k$, $$\operatorname{Spec}_\mathrm{Bal}(\operatorname{stab}(kG))\cong \operatorname{Proj}(H^\bullet(G,k)).$$

In 2013, Krause gave (pdf) a construction of $P(n,d)$ that more easily admitted a description of a monoidal structure.

In 2017, Aquilino and Reischuk showed (pdf) that the Schur-Weyl functor was monoidal!

If we are interested in computing the spectrum of $\mathrm{D}^\mathrm{b}(S(n,d))$, perhaps we can use the structure of $$\operatorname{Spec}_\mathrm{Bal}(\operatorname{stab}(k\mathfrak{S}_d))\cong \operatorname{Proj}(H^\bullet(\mathfrak{S}_d,k))$$ (which was proved first by Benson, Carlson, and Rickard: pdf) to say something about it, leveraging the monoidicity of the Schur-Weyl functor.

Besides the earlier references, the following can provide further directions for the interested reader.

- $q$-Schur Algebras
- Dipper, James — The $q$-Schur algebra
- Donkin — The $q$-Schur algebra
- Non-commutative spectra
- Nakano, Vashaw, Yakimov — Noncommutatuve tensor triangular geometry
- Nakano, Vashaw, Yakimov — Noncommutatuve tensor triangular geometry and the tensor product property for support maps