**Data**

- Title: Quivers for $\mathrm{SL}_{2}$ tilting modules
- Authors: Daniel Tubbenhauer and Paul Wedrich
- Status: Preprint. Last update: Fri, 26 Jul 2019 13:09:47 UTC
- ArXiv link: https://arxiv.org/abs/1907.11560

**Abstract**

Using diagrammatic methods, we define a
quiver algebra depending on a prime $\mathsf{p}$ and
show that it is the algebra underlying
the category of tilting modules for
$\mathrm{SL}_{2}$ in characteristic $\mathsf{p}$.
Along the way, we obtain a presentation for
morphisms between $\mathsf{p}$-Jones-Wenzl projectors.

**A few extra words**

Let $\mathbb{K}$ denote an algebraically closed field and $\mathbf{Tilt}=\mathbf{Tilt}\big(\mathrm{SL}_{2}(\mathbb{K})\big)$ the additive,
$\mathbb{K}$-linear category of (left-)tilting modules for the algebraic group $\mathrm{SL}_{2}=\mathrm{SL}_{2}(\mathbb{K})$.
This category can be described as the full subcategory of $\mathrm{SL}_{2}$-modules
which is monoidally generated by the vector representation $T(1)\cong\mathbb{K}^{2}$, and which is closed under
taking finite direct sums and direct summands.
The purpose of this paper is to give a generators and relations presentation of $\mathbf{Tilt}$
by identifying it with the category of projective modules for an explicitly described quiver algebra.
For $\mathbb{K}$ of characteristic zero this is trivial as $\mathbf{Tilt}$ is semisimple,
and the indecomposable tilting modules are indeed the simple modules. The quantum analog at
a complex root of unity is related to the zigzag algebra
with vertex set $\mathbb{N}$ and a starting condition.
The focus of this paper is on the case of positive characteristic $\mathsf{p}>0$, for which we represent
$\mathbf{Tilt}$ as a quotient $Z=Z_{\mathsf{p}}$ of the path algebra of an infinite, fractal-like quiver,
a truncation of which is illustrated for $\mathsf{p}=3$ as

This illustrates the full subquiver containing the first
$100$ vertices of the quiver underlying $Z_{3}$.

Note that the algebra $Z$ contains information about
the representation theory of $\mathrm{SL}_{2}$, as e.g. about the
Weyl factors $\Delta(w_{i}-1)$ in $T(v-1)$.
If the $\mathsf{p}$-adic expansion
$v=\sum_{i=0}^{j}a_{i}\mathsf{p}^{i}$
has exactly $r+1$ non-zero digits, then there are $2^{r}$ such factors and, correspondingly,
$r$ arrows from $v-1$ to certain $w_{i}-1

Note further the uniform behavior of $Z$ with respect to $\mathsf{p}$. For example

is (a cut-off) of the quivers $Z_{2}$, $Z_{5}$ and $Z_{7}$, which, zooming out
such that the precise labels get invisible, look basically the same as the one for $Z_3$.

The basis for our work is the classical fact that the Temperley-Lieb algebra controls the
finite-dimensional representation theory of $\mathrm{SL}_{2}$.
The second main ingredient is an explicit description
of $\mathsf{p}$-Jones--Wenzl projectors,
which are characteristic $\mathsf{p}$ analogs of the classical Jones-Wenzl projectors.
The bulk of this paper is devoted to a careful study
of morphisms between $\mathsf{p}$-Jones--Wenzl projectors over $\mathbb{F}_{\mathsf{p}}$
and the linear relations between them; a result of which we
think as being of independent interest.